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� Jungwirth, 1,2, * Rafael M. Fernandes, 3,4 Eduardo Fradkin, 3,4 Allan H. MacDonald, 5 Jairo Sinova, 6,7 Tomá s � 1,8,9, * and Libor Smejkal 1 Institute of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha, Czech Republic 2 School of Physics and Astronomy, University of Nottingham, NG7 2RD Nottingham, UK 3 Department of Physics, The Grainger College of Engineering, University of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA 4 Anthony J. Leggett Institute for Condensed Matter Theory, The Grainger College of Engineering, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA 5 Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA 6 Institut für Physik, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany 7 Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA 8 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany 9 Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany � *Correspondence: [email protected] (T.J.), [email protected] (L. S.) https://doi.org/10.1016/j.newton.2025.100162 The Pauli exclusion principle combined with interactions between fermions is a basic mechanism across condensed-matter systems giving rise to a spontaneous breaking of the spin-space rotation symmetry of spin-ordered phases. Ferromagnetism is a conventional manifestation of spin ordering that leads to numerous applications (e.g., in spintronic information technologies). Altermagnetism, whose recent discovery was largely motivated by spintronics, stands apart from conventional magnetism in the sense that it spontaneously breaks not only spin-space but also real-space rotation symmetries, while it pre serves a symmetry combining spin-space and real-space rotations. This is realized on crystals by a collinear compensated ordering of spins with a characteristic d-, g-, or i-wave symmetry. This perspective goes beyond the theory of spin arrangements on crystals by connecting altermagnetism to basic notions in condensed matter physics. Specifically, we reflect on the analogies and distinctions of altermagnetism as compared to superfluid 3 He and theories of spin ordering in the momentum space generated by other higher-partial-wave instabilities of a Fermi liquid. On one hand, all these physical systems have in common the extraordinary combination of spontaneous breaking of spin-space and real-space rotation symmetries. On the other hand, we point out that there are key differences, both at the symmetry level and, particularly, at the level of microscopic mechanisms of ordering. These explain the comparatively large abundance, robust ness, and utility of altermagnetism, as predicted by the symmetry classification of spin arrangements on crystals and ab initio calculations, and supported by initial experiments.
ACCESSIBLE OVERVIEW SUMMARY The Pauli exclusion principle combined with interactions between fermions is a basic mechanism across condensed-matter systems giving rise to a spontaneous breaking of the spin-space rotation symmetry of spin-ordered phases. Ferromagnetism is a conventional manifestation of spin ordering. Altermagnetism stands apart from conventional magnetism because it breaks not only spin-space but also real-space rota tion symmetries, while it preserves a symmetry combining spin-space and real-space rotations. This is real ized on crystals by a collinear compensated ordering of spins with a characteristic d-, g-, or i-wave symmetry. In this perspective, we connect altermagnetism to basic notions in condensed matter physics. Specifically, we reflect on the analogies and distinctions of altermagnetism as compared to superfluid 3 He and theories of spin ordering in the momentum space generated by other higher-partial-wave instabilities of a Fermi liquid. While these physical systems have in common the combination of spontaneous breaking of spin-space and real-space rotation symmetries, there are key differences both at the symmetry level and the level of Newton 1, August 4, 2025 © 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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microscopic mechanisms of ordering. These explain the comparatively large abundance, robustness, and utility of altermagnetism, as predicted by the symmetry classification of spin arrangements on crystals and ab initio calculations, and supported by initial experiments.
INTRODUCTION The research beyond conventional magnetism, which led to the recent delineation of the altermagnetic symmetry class, 1 was largely motivated by the field of spintronics. 2,3,4,5 From an applied perspective, spintronics is a modern branch of inte grated-circuit technologies undergoing a transition from niche to mass production, thanks in particular to embedded non-vola tile memories complementing semiconductors on advancenode processor chips. 6–8 The functionality of present spintronic memories is based on the magnetization in conventional ferro magnets that generates well separated and conserved spin-up and spin-down channels in the electronic structure (Figure 1A). Simultaneously, however, the magnetization sets physical limits on the spatial, temporal, and energy scalability of the spintronic technology. 2,3 Altermagnetism offers the prospect of removing these limits by combining well-separated and conserved spinup and spin-down channels with vanishing net magnetization (Figure 1B). 1,3 It enables this extraordinary combination of prop erties thanks to the unconventional anisotropic d-, g-, or i-wave nature of its magnetic ordering.
In this article, we dissect altermagnetic ordering from the sym metry and microscopic-mechanism perspectives, and connect it to basic condensed-matter physics notions developed in the research of instabilities in metallic Fermi liquids, namely of super fluid 3 He 9–14 and higher-partial-wave spin-channel Pomeran chuk instabilities. 15–28 Apart from analogies, we highlight key dis tinctions of altermagnetism to reflect on its comparatively large abundance, robustness, and foreseen utility in diverse science and technology fields. Indeed, altermagnetism was predicted to emerge in numerous materials, covering a broad range of interaction strengths and conduction types from weakly interact ing metals to strongly correlated Mott insulators, 1,3,4,29–31 and to enable interplay with semiconducting, superconducting, or ferroelectric phases. 1,3,32–40 Experimentally, altermagnetism has been observed in materials with ordering above room tem perature. 41–55
As with any spin-ordered phase, altermagnetism shares with superfluid 3 He and with the ordered phases generated by the spin-channel Pomeranchuk instabilities the spontaneously broken continuous spin-space rotation symmetry. 1,10,18,21,56–59 In addition, altermagnetism, superfluid 3 He, and the ordered phases generated by the higher-partial-wave spin-channel Pom eranchuk instabilities spontaneously break the real-space rota tion symmetry, and by doing so, they stand apart from conven tional magnetism. 1,9–22
In altermagnetism, the real-space symmetries concern discrete crystallographically constrained rotations (proper or improper and symmorphic or non-symmorphic). Remarkably, their spontaneous breaking in the ordered ground state can be prearranged by the crystal structure in the normal phase, which emphasizes the inherent role of the crystal lattice in altermagnet 2 Newton 1, August 4, 2025 ism. In contrast, 3 He is a uniform liquid of fermionic particles in a free space with no underlying lattice. In the normal phase, both the spin-space and the real-space rotation symmetries are thus continuous, and they spontaneously break, hand-in-hand, upon the transition to the ordered superfluid phase. 9–14 Similarly, the theory of Pomeranchuk instabilities commonly assumes a uniform Galilean-invariant Fermi liquid or, if included, the lattice effects enter only indirectly via non-Galilean anisotropic or non-quadratic corrections to the single-particle energy disper sion in the momentum space. 16,21,22
The inherent role of the crystal lattice in the altermagnetic ordering, and the absence of it in superfluid 3 He and the Pomer anchuk instability theories, underpin the principal distinction in the respective microscopic ordering mechanisms. In superfluid 3 He and in the spin-channel Pomeranchuk instabilities, the many-body interactions in the Hamiltonian of the fermionic sys tem, together with the Pauli exclusion principle, render the mo mentum-space Fermi surface of the metallic fluid unstable. In the superfluid 3 He case, and within a narrow window of low tem peratures and high pressures, this results in a spin-triplet Cooper pairing accompanied by the formation of an anisotropic quasi particle-excitation gap 9 (Figure 2A). In the case of the spin-chan nel higher-partial-wave Pomeranchuk instabilities, which so far have been experimentally elusive, the correlation-induced insta bility is predicted to lead to a spin-dependent anisotropic Fermi surface distortion 16,19,21,22 (Figure 2B). In contrast, the altermag netic ordering mechanism is applicable to both metallic and insulating systems. Here, the many-body Coulomb interactions and Pauli exclusion principle conspire with the single-particle crystal potential to generate an anisotropic spin density distribu tion in the crystal, reflected in the momentum space in the corre sponding spin-polarized anisotropically distorted energy isosurfaces (Figures 1B and 2C). 1,4
The predicted physical systems potentially hosting the spinchannel higher-partial-wave Pomeranchuk instabilities included heavy fermions, cold atoms, or 3 He. 21,22 The identified distinct mechanism of the altermagnetic ordering directed the experi mental search toward a different class of condensed-matter sys tems, 1,3 which has recently materialized in the spectroscopic and microscopic confirmation of altermagnetism in, among others, common binary compounds like MnTe or CrSb. 41–52
We conclude this opening section by brief comments on dipolar-coupling or spin-orbit-coupling terms originating from the relativistic Dirac equation, which explicitly couple the spinspace and the real-space degrees of freedom in the system’s Hamiltonian. As single-particle terms in the Hamiltonian, they are not responsible for the spontaneous ordering. They can, however, remove the degeneracy of different ordered ground states related by symmetries of the non-relativistic Hamiltonian. The resulting magnetic anisotropy energy can facilitate the sta bility of ordered ground states at discrete orientations of spins, which is, for example, the physical basis of the magnetic memory ll
functionality. Altermagnets can further enrich the landscape of the relativistic spin-dependent effects beyond the phenomenol ogy of conventional magnetic or non-magnetic systems. Exam ples are Hall responses, which are highly anisotropic and are not generated by internal magnetization or external magnetic field, non-linear altermagnetic relativistic spin-splitting effects in even-parity band structures, or novel topological phases. These phenomena, including their initial experimental exploration, are reviewed, for example, in references 2, 3, 31 , and 60–62 .
In the following sections, we focus on physics related to the ordering. For this, it is desirable to disentangle the spontaneous symmetry breaking in the ordered ground state due to manybody interactions, from the symmetry breaking in the non-inter acting single-particle Hamiltonian arising from the relativistic coupling of the spin-space and the real-space degrees of freedom. To do so, we consider symmetries of generally different transformations in the spin-space and in the real-space. This approach is analogous to symmetry theories earlier employed in the studies of superfluid 3 He and unconventional supercon ducting phases, 9–14,63–65 as well as of the Pomeranchuk Fermi liquid instabilities. 15–28 A spin-group theory, which systemati cally applies the approach of uncoupled spin-space and realspace to the symmetry classification of spin arrangements on crystals, 1,3,30,32,66–84 led to the recent delineation of altermag netism as a distinct third collinear magnetic phase, separate from conventional ferromagnetism and antiferromagnetism. 1
The discussion below is organized in the following sections. First, we introduce the symmetry classification and microscopic physics of altermagnetism. Second, we place ordered phases preserving the spin-space rotation symmetry in a two-parameter space of interaction strength and conduction type. The discus sion serves as a background reference for the follow-up sections on phases that spontaneously break both the spin-space and the real-space rotation symmetries. These two sections reflect on the analogies and distinctions of altermagnetism as compared to superfluid 3 He and ordered phases generated by the Pomeranchuk Fermi liquid instabilities, respectively. We then extend the comparison beyond the collinear altermagnetic ordering by including non-collinear compensated spin arrange ments on crystals. Finally, we briefly summarize our Perspective.
ALTERMAGNETISM The collinear altermagnetic ordering is exclusively and unambig uously delineated by the spin-group symmetries. 1 The non-rela tivistic many-body interacting Hamiltonian has the spin-group symmetry Z T 2 × SO(3) × G, where Z T 2 contains the time-reversal (T) symmetry, SO(3) is a group of all continuous spin-space rota tions, and G is the crystallographic point groups.
In the altermagnetically ordered ground state, the spin-
bines T with a 2-fold spin-space rotation C 2 around an axis orthogonal to the collinearity axis of spins, SO(2) is a group of continuous spin-space rotations around the collinearity axis, E is the spin-space identity, H is a halving subgroup of G, and Newton 1, August 4, 2025 3 ll OPEN ACCESS
G − H contains proper or improper crystal rotations and does not contain the parity (inversion) transformation. (Note that C 2 T 2 T Z C is not a normal 2 ⋉ SO(2) is a semidirect product since Z 2 C 2 T subgroup of Z
2 ⋉ SO(2).) The collinear altermagnetic ground state thus breaks the T sym metry, while it preserves the C 2 T symmetry. It also breaks the spinspace SO(3) symmetry, while preserving the SO(2) symmetry. Finally, the altermagnetic ground state breaks the symmetries of G − H, while it preserves the symmetries of H and symmetries combining transformations from G − H with the spin-space C 2 rotation. The latter symmetries protect the zero net magnetization of the compensated collinear altermagnetic ordering.
In summary, the altermagnetic spin groups explicitly describe the spontaneous breaking of both the spin-space and the realspace rotation symmetries. In contrast, collinear ferromagnetism is delineated by spin groups 1 Z 2 C 2 T ⋉ SO(2) × [E ‖ G], explicitly showing the preserved rotation symmetries of the parent crystal lographic point group G.
In the momentum space of the altermagnetic phase, the energy iso-surface of a given spin is distorted, breaking the symmetries of G − H and preserving the symmetries of H. The iso-surfaces cor responding to opposite spins are mutually related by the symme tries of G − H. Near the Γ-point, the electronic structure is spindegenerate at 2, 4, or 6 nodal surfaces in the three-dimensional 4 Newton 1, August 4, 2025 Figure 2. Ordered phases with sponta neous breaking of spin-space and realspace rotation symmetries (A) The A phase of superfluid 3 He with the aniso tropic Cooper-pairing (gap) function in the mo mentum space. Arrows represent the order
vector in the spin-space with one non-zero component given by the l z = 1 spherical harmonic expressed in the momentum space. (B) The α phase spin-channel l = 2 and l = 1 Pomeranchuk instabilities of a Fermi liquid, with arrows showing the momentum-dependent collinear spin polarization on the anisotropically distorted/shifted Fermi surfaces. (C) Representative altermagnetic and anti altermagnetic crystals with analogous mo mentum-space collinear spin polarization and energy iso-surface distortion/shift to the α phase Pomeranchuk instabilities. (D) The B phase of superfluid 3 He with the isotropic Cooper-pairing (gap) function in the momentum space. Arrows represent the order parameter spin-space vector, d B l = 1 (k)∝k, whose components are given by a combination of all three spherical harmonics l z = ± 1 and 0. (E) The β phase spin-channel l = 2 and l = 1 Pomeranchuk instabilities of a Fermi liquid with arrows showing the momentum-dependent spin texture on isotropic Fermi surfaces. (For brevity, we show only one of the two spin-split Fermi surfaces.) (F) Representative magnetic crystals with analo gous momentum-space spin textures to the β phase Pomeranchuk instabilities. In all panels, broken/preserved time-reversal (T) and inversion (parity P) symmetries in the momentum space are shown by ⨯/✔ marks.
(3D) Brillouin zone, depending on the spin group. On either side of the nodal surface, the sign of the spin-polarization alternates. Correspondingly, the altermagnetic ordering can be of d-, g-, or i-wave type. 1 Note that in the momentum space, the electronic structure has the parity symmetry, regardless of whether in the po sition space of the crystal lattice the parity symmetry is present or broken. This is due to the C 2 T symmetry, which, in combination with SO(2), acts as the parity symmetry. 1 (Here, the combined C 2 T transformation reverses momentum while keeping the spin invariant; in other words, it acts as the parity transformation.) We also note that the crystallographically constrained possible discrete real-space rotation transformations restrict the allowed even-parity-wave altermagnetism to d-, g-, or i-wave. 1
Numerous material candidates of not only d-wave but also g-wave and i-wave altermagnets have been identified using the spin-group classification, supported by density functional theory calculations. 1,3,29,30 The initial experimental spectroscopic confirmations were reported in g-wave altermagnets MnTe and CrSb, 41–52 followed by CoNb 4 Se 8 . 53 A spectroscopic evidence of d-wave altermagnetism was reported in KV 2 Se 2 O 54 or RbV 2 Te 2 O. 55
In the remaining part of this section, we leave the formal spin� group theory and, following Smejkal et al., 1,3 we dissect the alter magnetic symmetry breaking and microscopic ordering mecha nism using a model altermagnet. Here, we choose for the model ll
a square Lieb lattice. 85–87 Illustrations of the crystal structure in the position space and of the energy spectrum in the momentumspace are shown in Figure 3. In the normal phase, the model crystal has the continuous SO(3) spin-space symmetry and the crystallo graphic point group symmetry, from which we explicitly highlight the 4-fold rotation C 4 . These are symmetries of the whole crystal (unit cell) and, correspondingly, the electronic band structure in the normal phase is spin-degenerate and 4-fold symmetric in the momentum space. However, still in the normal phase, the C 4 sym metry is already locally broken on sites 1 and 2 (Figure 3A). The model energy spectrum shows two pairs of spin-degenerate bands, where one pair is dominated by orbitals from site 1 and the other pair by orbitals from site 2 (Figure 3A). The bands are anisotropic, reflecting the locally broken C 4 site symmetry, and the energy scale of their orbital splitting (E c in Figure 3A) is deter mined by the single-particle crystal potential.
The altermagnetic order spontaneously breaks the SO(3) spinspace symmetry. In addition, the C 4 symmetry, which was already broken in the normal phase locally on the two sites, be comes a spontaneously globally broken symmetry of the crystal unit cell in the altermagnetic phase. The ordered phase pre serves the symmetry combining the real-space C 4 rotation with the spin-space C 2 rotation. These symmetry features are high lighted by the schematic illustration of the anisotropic spin den sity and its decomposition into the dipole and the higher-partialwave (d-wave) components (Figure 3B).
Remarkably, the higher-partial-wave component of the spin density is ordered ferroically on sites 1 and 2. In analogy to ferroi cally ordered dipoles in conventional ferromagnetism (Figure 1A), this leads to broken time-reversal symmetry of the momentumspace electronic structure and to corresponding time-reversal symmetry breaking electronic responses. 4 The distinction from ferromagnetism is the higher-partial-wave nature of the ferroic component in altermagnetism, which corresponds to the higher-partial-wave symmetry of anisotropic exchange interac tions in the crystal lattice. 72 The spin polarization in the momentum-space electronic structure also reflects the higher-partial-wave (d-wave) symme try of the ferroic component of the anisotropic spin density in the position space of the crystal (Figure 3B). The spin-degeneracy of energy bands in the momentum-space is lifted in the altermag netic phase by the many-body interaction (E x exchange-energy scale in Figure 3B). Remarkably, however, the spin-splitting of nearby bands in the model band structure copies the anisotropic momentum-dependence and the magnitude of the orbital band splitting in the normal phase (E c crystal potential scale in Figures 3A and 3B). 1,3 This further illustrates that the sponta neous breaking of the global C 4 symmetry in the ordered phase is prearranged by the locally broken C 4 site symmetry in the normal phase.
The above link from the orbital splitting in Figure 3A to the spinsplitting in Figure 3B is an illustration of the microscopic mecha nism of the altermagnetic ordering in which the single-particle crystal potential conspires with the many-body electron-electron (exchange) interaction to form the unconventional phase sponta neously breaking both the spin-space and the real-space rota tion symmetries. The mechanism is robust thanks to the typical ∼eV scales of both the crystal potential and the electron-electron interactions, as confirmed in several altermagnetic candidate materials by density functional theory calculations. 1,3 (Note that for comparison, the relativistic dipolar or spin-orbit interac tions are typically on an ∼meV scale, unless involving heavyelement orbitals or extreme magnetic fields.)
Besides the robustness, altermagnetism is predicted to be abundant. Its microscopic ordering mechanism applies to both metallic and insulating systems. 1,3 In comparison, ferromagne tism is mostly realized in metals. Moreover, from the symmetry perspective, out of the 122/1,421 spin-point/spin-space groups of all collinear spin arrangements on 3D crystals, 37/422 corre spond to the altermagnetic order, in comparison to 32/230 cor responding to the conventional ferromagnetic order (and 53/ 769 to conventional antiferromagnetic order). 1 Newton 1, August 4, 2025 5 ll OPEN ACCESS
Figure 4. The landscape of other condensed-matter phases Horizontal axis corresponds to increasing strength of interactions. Insulating, metallic, and superconducting/superfluid phases are separated along the vertical axis. Bold symbols highlight phases with spontaneously broken spinspace rotation symmetry. cPI, charge Pomeranchuk instabilities; FM, ferro magnets; SC, superconductors; SF, superfluid; sPI, spin-channel Pomer anchuk instabilities.
ORDERED PHASES PRESERVING SPIN-SPACE ROTATION SYMMETRY We now turn to the comparison of altermagnetism with other condensed-matter phases, summarized in Figure 4. In this sec tion, we briefly recap, as a reference, phases that preserve the spin-space rotation symmetry. Painted with a broad brush, the following (non-exhaustive) classification by the conduction type and interaction strength can be made of phases without spin ordering, as shown in Figure 4. Starting from the bottomleft of the diagram, we have band insulators whose physics is described by an effective single-particle band picture featuring an energy gap separating completely filled bands from empty bands. The band-insulating phases are produced by quantuminterference effects of electrons in the periodic potential of the crystal lattice.
Metallic phases feature a Fermi surface separating the occu pied and empty electronic states. Landau’s Fermi liquid the ory 88,89 provides an elegant explanation why, in many metals, excited states near the Fermi surface can be represented by weakly interacting fermion quasiparticles whose lifetime be comes infinite when approaching the Fermi surface. This is because the scattering phase-space for the excited states near the Fermi surface is drastically limited by the Pauli exclusion principle. Normal metals falling into this weak-interaction Fermi liquid regime are depicted in the middle-left part of the diagram in Figure 4.
At sufficiently low temperatures, an arbitrarily weak attractive interaction (e.g., mediated by phonons) between the quasiparti cles near the Fermi surface leads to the formation of Cooper pairs of opposite-spin and opposite-momentum quasiparticles, corresponding to a spin-singlet (S = 0) s-wave (l = 0) pairing (Figure 5A). The occupied and empty fermionic states are sepa rated by an excitation gap when the Cooper pairs condense into the conventional Bardeen-Cooper-Schrieffer (BCS) supercon 6 Newton 1, August 4, 2025
ducting state, 90 which we correspondingly placed in the topleft portion of Figure 4.
The energy iso-surfaces in the band insulators and the normal metals, or the Cooper-pairing order parameter in the s-wave superconductors (Figure 5A), preserve the spin-space and the real-space rotation symmetries. Note that the superconduct ing/superfluid ordering spontaneously breaks the U(1) gauge (particle-conservation) symmetry.
We now move to the unconventional superconductivity 11,64 in the strongly interacting part of the diagram in Figure 4. Because of strong short-range repulsive interactions, electrons in the Cooper pairs favor anisotropic pairing with relative angular mo mentum l > 0. This breaks the real-space rotation symmetry. Un conventional d-wave (l = 2) cuprate superconductors belong to this class (Figure 5A). The symmetric orbital part of the pairing function dictates, by the Pauli exclusion principle, an antisym metric spin-singlet (S = 0) part of the pairing function. The d-wave superconducting phase, like the conventional s-wave superconductivity, thus preserves the spin-space rotation symmetry.
Wigner crystal is another ordered phase in the strongly inter acting part of the diagram in Figure 4. It is stabilized when the interaction energy dominates the kinetic energy at sufficiently low electronic densities. Ordering in this insulating phase is char acterized by a non-uniformity (i.e., by a spontaneous breaking of the translation symmetry). Charge-density waves, stripe states, or electronic smectic liquid crystals are other examples of corre lated non-uniform ordered phases. 91 They can be insulating or metallic.
Finally, we arrive at the central part of the diagram in Figure 4 corresponding to an intricate metallic intermediate-interaction regime. Coming from the right, the corresponding ordered phases can be viewed as melted charge-density waves, 91–95 recovering the translation symmetry but not the real-space rota tion symmetry. 91 Coming from the left, they can be described as Pomeranchuk Fermi liquid instabilities in the l > 0 charge chan nel, characterized by distortions of the Fermi surface and the corresponding spontaneous breaking of the real-space rotation symmetry (Figure 5B). 19,91,96 Simultaneously, they preserve the uniformity (translation symmetry) of the Fermi liquid and the spin-space rotation symmetry.
We elaborate on the framework of the Pomeranchuk Fermi liquid instabilities in the next section, where we move to the discussion of the spin-ordered phases. 16,19–22,25–28 Here, we conclude the overview of the phases preserving the spinspace rotation symmetry by pointing out that the chargechannel l = 1 Pomeranchuk instability has been the subject of an ongoing discussion throughout the past 100 years. In Figure 5B, the instability is depicted as a Fermi surface distor tion in the form of a parity-breaking shift. Such a Fermi surface distortion suggests a presence of a spontaneous equilibrium current. From the 1920s to the 1940s, recurring attempts were made by Bloch, Landau, and Born, 97–100 among others, to explain conventional superconductivity by this instability. However, the theory failed as it violated the first theorem on superconductivity, formulated by Bloch himself, showing that the minimum energy state bears no current. 99 With the other unsuccessful theories at the time, Bloch was led to his ll
‘‘second theorem,’’ which states that every theory of super conductivity can be disproved. 99 While conventional super conductivity was eventually explained by the Cooper-pairing mechanism, 90 whether or not the charge-channel l = 1 Pom eranchuk instability is physically possible has remained a mat ter of theoretical research up to today. 26–28 Experimentally, it has remained elusive.
The charge-channel l = 2 Pomeranchuk instability, leading to an ordered phase referred to as nematic, 19,91,96 has a form of an even-parity anisotropic Fermi surface distortion (Figure 5B). The studied physical realizations of this electronic nematic phase include semiconducting 2D electron systems at high mag netic fields in the vicinity of correlated fractional quantum Hall states, correlated ruthenates at high magnetic fields in the vicin ity of a metamagnetic transition, and unconventional supercon ductors such as cuprates and iron pnictides. 19,101–106
Finally, we note that no general limitations have been identified to exclude l > 2 Pomeranchuk instabilities. For example, a 5-fold l = 5 Fermi surface distortion has been theoretically considered in a uniform Fermi liquid unconstrained by crystal lattice symmetries. 107 SUPERFLUID 3 HE AND ALTERMAGNETISM We now turn the focus to the ordered phases that spontaneously break both the spin-space and the real-space rotation symme tries and compare them to altermagnetism. Starting from the top of the diagram in Figure 4, the well-established representa tive 12 is superfluid 3 He. Since the late 1950s, more than a decade before the experimental discovery, theorists were considering extensions of the BCS theory to the case of charge-neutral spin-1/2 3 He atoms. Here, the longer-range attractive van der Waals interaction and the strong short-range repulsion, comple mented by the additional effective interaction due to spin fluctu ations, favor spin-triplet p-wave pairing (S = l = 1). 12
Despite the p-wave pairing, the expectation was that 3 He would remain essentially isotropic in the superfluid phase. Specifically, the lowest energy superfluid state originally predicted by the micro scopic theory, the so-called B phase, is formed by a condensation of Cooper pairs given by a linear combination containing all three l z
S z ′ = 0〉. (We recall that dipolar and spin-orbit coupling is omitted, which we highlight in the pairing by introducing distinct, unprimed, and primed coordinate systems for the real space and the spin space.) Using l = 1, l z = ± 1; 0 spherical harmonics, the B phase order parameter along the Fermi surface is commonly written as a vector in the spin-space whose components depend on the mo mentum unit vector as 9–14 d B l = 1 (k)∝k. The direction of d B l = 1 (k) varies with k, but the constant amplitude implies that the quasipar ticle excitation gap in the B phase is open along the whole Fermi surface and has a constant value (i.e., it is isotropic) (Figure 2D).
We emphasize that the order parameter vectors d l = 1 (k) describing superfluid phases of 3 He, albeit not directly propor tional to spin, are constructed to follow the same rotation as the spin quantization axis under spin-space rotations. 9–14 The order parameter d B l = 1 (k) thus explicitly shows that the B phase of superfluid 3 He breaks individually the spin-space rotation symmetry SO(3) s and the real-space rotation symmetry SO(3) r . Simultaneously, it retains an SO(3) symmetry of the same com bined rotations in the spin-space and the real-space, underlying the essentially isotropic nature of the B phase (Figure 2D). In addition, d B l = 1 (k) shows that the B phase retains the timereversal symmetry and breaks the parity symmetry (Figure 2D).
Consistent with the original theoretical expectation, the experi mental phase diagram of superfluid 3 He is indeed dominated by the B phase. A major surprise thus was the experimental observa tion of the A phase of superfluid 3 He, occurring within a narrow win dow of low temperatures and high pressures. 12,108 Its spintriplet p-wave Cooper-pairing function is of the form |l z = 1; S z ′ = 1〉 + |l z = 1;S z ′ = − 1〉, containing only one of the three l z values. The corresponding order parameter is given by d A l = 1 (k)∝(0; ik x − k y ; 0). For all momenta, the vector has only one non-zero component (i.e., it is collinear along the Fermi sur face). Its amplitude, however, varies with momentum, rendering the A phase anisotropic, with nodes in the quasiparticle excitation gap at k x = k y = 0 (Figure 2A). There is no remaining SO(3) symme try in the A phase, which emphasizes its anisotropic nature. The or der parameter d A l = 1 (k), containing only one l z component, also im plies broken time-reversal symmetry in the A phase (Figure 2A).
The nodes in the gap function make the A phase of superfluid 3 He less favorable than the B phase with the isotropic nodeless gap function. The surprising and limited occurrence of the aniso tropic A phase of superfluid 3 He is then explained by a subtle feedback effect of the pairing state on the spin fluctuations that contribute to the attraction forming the Cooper pairs. 109 The B phase has a reduced spin susceptibility compared to the normal phase and the A phase. Since the pairing mechanism involves spin fluctuations, this can disfavor the B phase. The Newton 1, August 4, 2025 7 ll OPEN ACCESS lesson learned from superfluid 3 He was that it may require an intricate and subtle interplay of microscopic interactions to realize an ordered phase that spontaneously breaks both the spin-space and real-space rotation symmetries and where these lead to an anisotropic nodal character of the ordering.
Altermagnets represent a new physical realization of the ordered phase belonging to the symmetry class with spontaneously broken spin-space and real-space rotation symmetries. It also shares with the A phase of superfluid 3 He the anisotropic nodal character (Figures 2A and 2C). In contrast to 3 He, however, the altermagnetic ordering was theoretically anticipated prior to the experimental discovery. 3 Moreover, the expectation was based not only on a theory applied to a specific physical system but also on the general spin-group classification of spin arrangements on crystals. 1 As a result, numerous altermagnetic candidates have been identified, many of which order at ambient conditions, and not only in 3D inor ganic materials 1,3,29,30 but also in 2D 3,71,85,86,110–114 and organic crystals. 115,116 The initial experimental demonstrations by mo mentum-space spectroscopic measurements have been per formed in room temperature altermagnetic materials MnTe and CrSb, 41–52 representing simple binary compounds readily avail able in stable high-quality bulk or thin-film forms. The spectros copy has been complemented in MnTe by position-space vector imaging and control of the altermagnetic ordering from micronscale single-domain states to nanoscale domain walls and topo logical vortices. 46
The robustness of altermagnetism stems from the specific microscopic physics of ordering. For superfluid 3 He, we mentioned above the key role of the subtle effects of the effective attractive interaction via spin fluctuations. The effective interac tions and, in general, the vicinity of other (fluctuating) phases of the interacting Fermi fluid, 117 have been a common theme considered across the field of the anisotropic phases, including the unconventional superconductors and the charge-channel Fermi liquid instabilities, discussed in the previous section. Alter magnets stand apart here. Although intriguing physics may arise from an interplay of the altermagnetic phase with other order pa rameters of the interacting electrons, 3,31 altermagnetic ordering is not based on this interplay. As described above, altermagnet ism is primarily stabilized, apart from the many-body exchange interaction, by the static single-particle potential of the underly ing crystal lattice. 1,3,4,31 The crystal potential energy scale tends to be strong, which adds to the robustness of altermagnets in metallic and insulating materials over a broad range of elec tron-electron interaction strengths.
SPIN-CHANNEL POMERANCHUK INSTABILITIES AND ALTERMAGNETISM In this section we compare altermagnetism to theoretically stud ied spin-ordered phases generated by spin-channel Pomeran chuk instabilities of the momentum-space Fermi surface. They are placed in the part of the diagram in Figure 4 corresponding to metallic conduction and intermediate interaction strengths. Together with the charge-channel Pomeranchuk instabilities mentioned earlier, their phenomenological description can be placed under the common umbrella based on Landau Fermi liquid theory. The theory considers two-body interactions be 8 Newton 1, August 4, 2025
tween the quasiparticles. In a free space and for quasiparticles near the Fermi surface, the two-body interaction depends only on the angle θ between the linear momenta of the quasipar ticles. Accordingly, it can be expanded in series of (l, l z = 0) spherical harmonics, correspondingly called angular-momentum l-channels. 88,89 The l = 0 component corresponds to an isotropic interaction, while the l > 0 components describe anisotropic inter actions. The prefactors of the expansion are the phenomenolog c(s) ical Landau parameters, F
l . Here, the superscript c(s) labels the charge (spin) channel interaction given by the sum (difference) of interactions of the same-spin and opposite-spin quasiparticles.
Pomeranchuk derived a general form of the static susceptibil ity in each charge, spin and l-channel, finding it to be propor
channel l = 0 (s-wave) instability corresponds to the conven tional metallic ferromagnetic phase. 88,118 The anisotropically distorted spin-degenerate Fermi surfaces, mentioned earlier and illustrated in Figure 5B, correspond to the charge-channel l = 1 (p-wave) 88,97–99,118 and l = 2 (d-wave) 19,91,96 Pomeranchuk instabilities, respectively.
The spin-channel l = 1 and l = 2 counterparts are illustrated in Figures 2B and 2E. Starting from l = 1, referred to as spin nematic in the early literature, 17,18 and later identified as a gener alized electronic liquid crystal, 22 there are two types labeled in references. 21 and 22 as the α phase and the β phase (Figures 2B and 2E), in analogy to the A phase and the B phase of superfluid 3 He (Figures 2A and 2D).
In the β phase, the Fermi surface spin-splits into larger and smaller surfaces, where the shape of each surface is otherwise undistorted. 15,17,21,22 This is reminiscent of the l = 0 (s-wave) ferromagnetic instability, but there are key symmetry differ ences. First, the β phase l = 1 Pomeranchuk instability shares with the B phase of superfluid 3 He the time-reversal symmetry and the broken parity symmetry. In contrast, the ferromagnetic instability breaks time-reversal symmetry and preserves the parity symmetry. Second, in analogy to the spin-space order parameter vector d B (k)∝k in the B phase of superfluid 3 He, the l = 1 β phase has a momentum-dependent spin texture on the energy iso-surface, where the direction of spin depends on the direction of momentum such that the spin winds once along the Fermi surface. As a result, the l = 1 β phase breaks, individually, both the spin-space and real-space rotation sym metries, but it preserves a symmetry combining the same rota tion transformations in the spin-space and the real-space, rendering the phase essentially isotropic. This is analogous to the B phase of superfluid 3 He. In contrast, the l = 0 ferromag netic Pomeranchuk instability is isotropic because it only breaks the spin-space rotation symmetry but preserves the real-space rotation symmetry.
The α phase generated by the l = 1 spin-channel Pomeran chuk instability is illustrated in Figure 2B as spin-split Fermi sur faces shifted along one direction for one spin and the opposite direction for the opposite spin. 16,17,21,22 The phase is more remi niscent of the A phase of superfluid 3 He in the following sense. Besides breaking the spin-space and real-space rotation sym metries, and in analogy to d A l = 1 (k)∝(0; ik x − k y ; 0) in the A phase ll
of superfluid 3 He, spins in the α phase are collinear on the Fermi surfaces, and the phase is anisotropic with spin-degenerate no des. However, the l = 1 α phase Pomeranchuk instability, like the l = 1 β phase but unlike the A phase of superfluid 3 He, retains the time-reversal symmetry.
In the spin-channel Pomeranchuk instabilities, the timereversal symmetry is broken for spin-ordered phases gener ated by the l = 2 (even l) instability (i.e., in phases retaining the parity symmetry). 19,22 The α phase l = 1 and l = 2 Pomeran chuk instabilities share with the A phase of superfluid 3 He the broken spin-space and real-space rotation symmetries and the anisotropic collinear and nodal character of ordering (Figures 2A and 2B). However, they depart from the A phase of superfluid 3 He in one of the two discrete symmetries (Figures 2A and 2B). The A phase of superfluid 3 He spontane ously breaks both the time-reversal and parity symmetry, while the l = 1 α phase Pomeranchuk instability retains the timereversal symmetry and breaks the parity symmetry. Vice versa, the l = 2 α phase Pomeranchuk instability breaks the timereversal symmetry and retains the parity symmetry. (Recall here that the even-parity superfluid/superconducting phases with even-l Cooper-pairing are spin-singlet and do not break the spin-space rotation symmetry.)
An illustration of momentum-space spin-dependent energy iso-surfaces of a d-wave altermagnet (Figure 2C) outlines the symmetry analogy to the l = 2 α phase Pomeranchuk instability (Figure 2B). However, the microscopic mechanism that stabi lizes the altermagnetic order in a broad range of materials, including insulators, is principally distinct from the Pomeranchuk instabilities. In the latter case, the interactions must be strong enough for the corresponding Landau parameter F l s to reach the critical value of the Pomeranchuk instability in the given l-channel, while, simultaneously, the interactions must be finetuned to avoid the instability in another l-channel. In particular, eliminating a pre-emptive l = 0 ferromagnetic instability was recognized as one of the key challenges for realizing the spin-or dered phases by l > 0 Pomeranchuk instabilities. 16,21,22,25 Apart from the isotropic ferromagnetic ordering, a spin density-wave phase can compete with the l > 0 Pomeranchuk instabilities from the non-uniform side 16 of the diagram in Figure 4. To date, the spin-channel l > 0 Pomeranchuk instabilities (without an interplay of strong spin-orbit coupling) have remained largely theoretical concepts.
We also point out that in theories of the spin-channel Pomer anchuk instabilities, the crystal lattice tends to enter only indi rectly via a modification of the one-body momentum-dependent energy dispersion of the quasiparticle states. For example, an anisotropic quasiparticle-dispersion and corresponding Fermi surface nesting effects in suitable lattice models were consid ered a possible mechanism for favoring the l = 1 α phase insta bility against the l = 0 ferromagnetic instability. 16 Alternatively, effects beyond quadratic quasiparticle dispersion were shown to determine whether, for a critical value of a given Landau parameter F l s > 0 , the corresponding ordering will tend to be of the α phase or the β phase. 22
In contrast, the crystal lattice plays an inherent role in the alter magnetic ordering. As a result, the altermagnetic spin-group symmetry protects zero net magnetization (i.e., excludes a ferro OPEN ACCESS magnetic component). Moreover, the collinearity of the spin arrangement in altermagnets in both the position space of the crystal lattice and the reciprocal momentum space excludes the spin-textured type of ordering, which in the Pomeranchuk in stabilities corresponds to the β phase.
To further highlight the distinction between microscopic ordering mechanisms of altermagnetism and the Pomeranchuk instabilities, we compare minimal models associated with each scenario. We focus for concreteness on d x 2 − y 2 -wave symme try. In the Pomeranchuk scenario, the electronic dispersion in the spin-ordered state near the Γ point becomes E σ (k) = ∕ E 0 (k) + ϕ(k x 2 − k y 2 )σ, where σ = ± 1 is the spin index, ϕ = 0 in the ordered state, and E 0 (k) ∼ k 2 . For this ordering transition to take place, the Landau parameter F 2 s must overcome a their critical threshold value, while all F s = 2 must remain below l∕ values.
This scenario, however, is very challenging to realize in a microscopic model. We can illustrate it on electrons hopping on a simple square lattice (Figures 1A and 1C) with an onsite (Hubbard) repulsion U. While the crystal potential modifies the non-interacting energy dispersion E 0 (k), the latter has the same form as the one introduced above close enough to the Γ point. The interaction U may drive a magnetic instability, whose nature depends on the model’s parameters. For instance, if the dispersion is such that the density of states has a strong peak at the Fermi level (e.g., a van Hove singularity), then a ferromag netic state may emerge through the Stoner mechanism for small enough U, leading to the l = 0 spin-channel Pomeranchuk insta bility. However, a weak-coupling instability toward an antiferro magnetic Néel state is possible if the dispersion has nesting features. The antiferromagnetic state also emerges in the strong-coupling regime at half-filling inside the Mott insulating state. Thus, an l = 2 spin-channel Pomeranchuk instability likely requires other types of interactions.
The altermagnetic ordering mechanism is fundamentally different, as here the crystal potential plays an essential role. We have illustrated this in the model square Lieb lattice in Figures 1B and 3. In contrast to the simple square lattice analyzed above, the next-nearest-neighbor hopping parameters are different along the x and y directions, manifested in a hop ping anisotropy t a . 87 The onsite Hubbard repulsion U can still drive an instability toward an antiparallel alignment of the spins. 119 Expanding the electronic dispersion around the Γ point,
ergy scale and t is the nearest-neighbor hopping. The dispersion has the same form as the electronic dispersion of the ordered state generated by the l = 2 spin-channel Pomeranchuk insta bility. However, in the altermagnetic ordering, the prefactor of the second term is a product Nt a , instead of the single parameter ϕ in the Pomeranchuk instability case. 119 This illustrates that both the effects of the single-particle crystal potential (encoded here in t a ) and of the many-body interactions (encoded in N) are required to yield an altermagnetic state. In contrast, in the case of the Pomeranchuk instability, there is a single and purely electronic energy scale (encoded in ϕ).
Importantly, for a given crystal potential, there are different known routes by which interactions can drive an instability toward an antiparallel configuration of spins. For instance, Newton 1, August 4, 2025 9 ll OPEN ACCESS besides the weak-coupling scenario outlined here, a strongcoupling Mott insulating state could host the same spin config uration. This contributes to the much larger versatility of altermagnetism as compared to the Pomeranchuk instability scenario, since the former can be realized in metals, insulators, and semiconductors, whereas the latter is only realized in metals.
Finally, we point out that the more detailed theoretical explora tion of the spin-channel Pomeranchuk instabilities was largely motivated by spintronics, 21 a pattern that was repeated 15 years later in altermagnets. 2,3,4 Also here, however, there is a significant difference. A detailed exploration of the spin-chan nel l = 1 β phase Pomeranchuk instability followed shortly after the theoretical predictions of the spin Hall charge-to-spin con version phenomenon in non-magnetic systems, generated by the relativistic spin-orbit coupling. 120,121 The time-reversal invariant spin texture in the l = 1 β phase instability was called a dynamically generated spin-orbit coupling because of the resemblance to the relativistic spin texture. 21 It was emphasized that compared to the perturbatively weak relativistic spin-orbit coupling proportional to 1=c 2 , where c is the speed of light, the dynamically generated spin-orbit coupling could lead to a signif icantly larger charge-to-spin conversion efficiency. However, an experimental realization of the spin-channel l = 1 β phase (or α phase) Pomeranchuk instability has remained elusive.
The search for altermagnets was motivated by a different spin tronics incentive. It followed after several years of intense research of spintronics based on collinear antiferromagnets with spin-degenerate band structures. 122 The driving idea was to leverage the superior spatial, temporal, and energy scalability demonstrated in antiferromagnetic spintronic devices, stem ming from the compensated magnetic ordering, while having well-separated and conserved spin-up and spin-down channels that underpin the technologically successful spintronics based on ferromagnets. 2,3,4 As reviewed in references 2 and 3, such a combination of merits, traditionally considered mutually exclu sive, is enabled by the altermagnetic ordering. Besides spin tronics, the extraordinary nature of altermagnetism is projected to be favorable in a range of research fields, from topological magnetism to hybrid systems integrating altermagnetism with semiconducting, superconducting, or ferroelectric phases, as reviewed in references 2, 3, and 31 .
SPIN ARRANGEMENTS ON CRYSTALS BEYOND ALTERMAGNETISM In the previous section, we left aside an apparent conflict be tween the theoretically considered ordered phases generated by the l = 1 Pomeranchuk instabilities, and the theorem by Bloch, which states that the minimum energy state bears no cur rent. 99 The diverging susceptibilities at critical values of the c(s) Landau Fermi liquid parameters, F
l = − 1, were identified by Pomeranchuk in all angular-momentum channels, including l = 1. This seems to violate the above theorem. Before discus sing spin arrangements on crystals beyond the even-parity-wave altermagnets, in which p-wave magnetism has been recently predicted, 75,76 it is thus desirable to first revisit this apparent conflict in the theory of Fermi liquid instabilities.
10 Newton 1, August 4, 2025
The divergences of the charge and spin-channel sus ceptibilities for l = 1 order parameters were analyzed in refer ences 26–28 . The studies started from the exact expression for the static susceptibility for a generic order parameter. It goes c(s) beyond the Pomeranchuk ∼ 1=(1 + F
l ) form by including quasiparticle states away from the Fermi surface. 123 For l = 1 or der parameters corresponding to the current of conserved charge or spin, the divergencies in the generalized susceptibil ities indeed disappear, regardless of the quasiparticle dispersion (i.e., both for the free-space Fermi liquid and in the presence of a crystal lattice). 26,27 This is consistent with the above theorem by Bloch.
However, two possibilities were identified that could enable the ordering phase transitions by the charge or spin-channel l = 1 Pomeranchuk instabilities. One is a generic form of the l = 1 order parameter, which does not correspond to the charge or spin current. 27 The other one follows from the obser vation that even in the cases where the static susceptibility is non-diverging for the charge or spin-current order parameter, the instability of the Fermi liquid ground state can still be signaled by the dynamic susceptibility. 28 In conclusion, present theories do not exclude the ordered l = 1 phases, even within the framework of the momentum-space Fermi liquid instabil ities. Their real physical realizations have, however, remained elusive.
This brings us to the recent studies of material realizations of odd-parity-wave magnetism (Figure 1D). 75,76,83,124–126,127–129 The characteristic symmetry of the corresponding spin ar rangements on crystals is time reversal combined with transla tion, which can be realized by an antiferroic ordering of local atomic dipoles in the crystal lattice. This symmetry is absent in altermagnetism, while it is reminiscent of conventional collinear Néel antiferromagnetism (Figure 1C). However, in addition to the symmetry combining time reversal with transla tion, the spin arrangement on the crystal lattice shown in Figure 1D is non-collinear coplanar, and spontaneously breaks parity symmetry. The non-collinearity can originate from, for example, frustrated exchange interactions on the crystal lattice even in the absence of spin-orbit coupling or from Dzyaloshinskii-Moriya interaction. (Even in the case where spin-orbit coupling contributes to the stabilization of the noncollinear spin ordering, the spin-ordering symmetry can be described by spin groups.) As a result, on one hand, the elec tronic structure in the momentum space has time-reversal sym metry, like in the collinear Néel antiferromagnetism, since a po sition-space translation does not change the momentum. On the other hand, it can show a nodal higher-partial-wave ordering in the momentum space with an alternating sign of a collinear spin polarization, and with both the spin-space and the real-space rotation symmetries broken, as in altermagnet ism. Hence, we label this odd-parity phase ‘‘antialtermagnetic,’’ and we show a p-wave example 75 in Figure 1D. In the momentum space, it has analogous symmetries to the α phase l = 1 Pomeranchuk instability (Figures 2B and 2C).
The momentum-space spin textures, analogous to the β phase l = 1 Pomeranchuk instability, can be realized on crystals with a non-coplanar instead of a coplanar spin arrangement, while keeping the symmetry combining time reversal with translation, ll
and the broken parity symmetry. 76,127 This is shown in Figures 2E and 2F. Finally, in Figure 2F, we also illustrate an example of noncollinear coplanar spin arrangements on crystals without the symmetry combining time reversal and translation, 2,5,130–133 and with a symmetry combining spin-space and real-space rota tions, resulting in a momentum-space spin texture analogous to the β phase l = 2 Pomeranchuk instability (Figure 2E).
CONCLUSION In this Perspective, we have discussed the unconventional spin ordering of the altermagnetic phase. The extraordinary nature of altermagnetism is that it spontaneously breaks the continuous spin-space rotation symmetry and the discrete real-space rota tion symmetry of the crystallographic point group, while preser ving a symmetry combining rotation transformations in the spinspace and the real-space. We have discussed key distinctions in symmetry and in microscopic ordering mechanisms between altermagnetism and ordered phases generated by momentumspace instabilities of a Fermi liquid to shed light on the abun dance and robustness of altermagnetism. This can be summa rized in the following points.
First, the A phase of superfluid 3 He was an experimental sur prise, and its stability is limited to a narrow range of low temper atures and high pressures. Material realizations of the α phase Fermi liquid instabilities (as well as other spin-channel l > 0 Pom eranchuk instabilities) have remained elusive. In contrast, alter magnetism was predicted by the systematic spin-symmetry group classification in a large family of materials and confirmed by microscopic density functional theory calculations. The the ory has guided the initial experimental verifications in room tem perature altermagnets.
Second, the narrow stability range of the A phase of superfluid 3 He was ascribed to the interplay of internal interactions in the 3 He Fermi fluid, namely the short-range repulsion, the long-range attractive van der Waals interaction, and the effective pairingdependent interaction due to spin fluctuations. Altermagnetism is distinct in that the ordering is stabilized by the internal elec tron-electron (exchange) interaction, together with the external single-particle potential of the static crystal lattice. In altermag netism, the crystal lattice plays an inherent part in the robust microscopic ordering mechanism. In contrast, 3 He is a homoge neous fluid with no underlying crystal lattice. In the theories of Pomeranchuk Fermi liquid instabilities, the crystal potential, if considered, enters only indirectly via a modified single-particle energy dispersion in the momentum space.
Third, ordered phases generated by the higher-partial-wave Pomeranchuk instabilities of a Fermi liquid are predicted to require interaction strength exceeding the critical value for the l > 0 spin-channel instability, while avoiding the conventional l = 0 ferromagnetic instability. In altermagnetism, the micro scopic ordering mechanism is effective from weak to strong interaction regimes, and a ferromagnetic component is excluded by the spin-group symmetry.
As a result of these distinctive features, altermagnetism is abundant and robust, which suggests that its unconventional ordering can open fruitful research directions in both science and technology.
ACKNOWLEDGMENTS We acknowledge helpful comments and suggestions on the manuscript by Anthony Leggett. We acknowledge Rodrigo Jeaschke and Anna Birk Hellenes for sharing with us unpublished results. T.J. acknowledges support from ERC Advanced Grant 101095925 and the Ministry of Education of the Czech Re public grant CZ.02.01.01/00/22008/0004594; R.M.F. acknowledges support from the Air Force Office of Scientific Research under award no. FA9550-21- 1-0423; E.F. acknowledges support from the US National Science Foundation grant DMR 2225920 at the University of Illinois; A.H.M. acknowledges support from the the NSF through the Center for Dynamics and Cntrol of Materials: an NSF Materials Research Science and Engineering Center under cooperative � acknowledge support agreement No. DMR-2308817 ; and J.S. and L. S. from Deutsche Forschungsgemeinschaft (German Research Foundation [DFG]), Project 452301518, and TRR 288-422213477 (project A09). L. � S. also acknowledges support from the ERC Starting Grant 101165122.
AUTHOR CONTRIBUTIONS Writing—original draft, T.J.; writing—review & editing, T.J., R.M.F., E.F., A.H. S. Lattice model at the bottom of Figure 2F—L. � S. M., J.S., and L. � DECLARATION OF INTERESTS J.S. is an advisory board member of Newton.
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