Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry

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

Popular Summary

Water and ice are classic examples of distinct phases of matter. They also serve as a classic example for illustrating symmetry principles for describing the distinct phases. In the case of water and ice, it is the presence or absence of rotational symmetry in physical space that determines the phase. Similarly, nonmagnetic and magnetic phases are distinguished by the presence or absence of a rotational symmetry, but this time in “spin space.” Traditionally, the landscape of magnetic phases is divided into two basic parts: ferromagnetism and antiferromagnetism. In our work, we develop a spin symmetry to describe a so-far overlooked third magnetic phase that we dub altermagnetism.

Ferromagnets are characterized by an uncompensated spin order in the momentum space of electrons, with nondegenerate spin-up and spin-down electronic channels resulting in nonzero net magnetization. Antiferromagnets have no spin order in momentum space, which makes their detection and control notoriously difficult. The newly identified altermagnetism is characterized by a spin order in the electronic momentum space that, however, is compensated. Specifically, the spin-up and spin-down channels in altermagnets are nondegenerate but equally populated and anisotropic. They are mutually connected by symmetry, combining spin space and physical space rotations.

Remarkably, altermagnetism allows for a realization of a long-sought but elusive magnetic counterpart of unconventional “$d$-wave superconductivity.” This opens a new fundamental research front of magnetism and many-body quantum phases. For spintronics, altermagnets share with antiferromagnets high spatial and temporal scalability, and with ferromagnets a range of efficient means to detect and control their magnetic state.

Figure 1

Illustration (in columns) of the three nonrelativistic collinear magnetic phases. Top box: Illustrative collinear spin arrangements and magnetization densities on crystals. Opposite spin directions are depicted by blue and red color. Spin arrows are placed outside the real-space cartoons to highlight that the overall spin axis orientation is not related to the real space coordinates for the nonrelativistic spin-group symmetries. 1st magnetic phase (conventional ferromagnetism) crystal corresponds to Fe, 2nd magnetic phase (conventional antiferromagnetism) to MnPt, and the 3rd unconventional magnetic phase (altermagnetism) to ${\mathrm{RuO}}_2$. Magenta arrow and magenta label highlight opposite-spin-sublattice transformation symmetries characteristic of the 2nd magnetic phase (real-space translation or inversion) and the 3rd magnetic phase (real-space rotation). Bottom box: Cartoons of band-structures and corresponding energy iso-surfaces show ferromagnetically spin-split bands (opposite spin states depicted by blue and red color), a spin-degenerate antiferromagnetic band, and bands in the 3rd magnetic phase with alternating sign of the spin splitting. The opposite-spinsublattice transformation of the spin Laue group which maps the same-energy eigenstates with opposite spins on the same $\mathbf{k}$-vector in the 2nd magnetic phase and on different $\mathbf{k}$-vectors in the 3rd magnetic phase is again highlighted. The remaining rows give the spin Laue group structure for the given phase with the number of different groups in brackets, and presence/absence of time-reversal-symmetry breaking, compensation and $d$-,$g$-, and $i$-wave symmetries.

Figure 2

Classification of spin-momentum locking in the third magnetic phase protected by spin-group symmetries, and material candidates. The columns describe the characteristic planar ($P$) or bulk ($B$) spin-momentum locking on model Hamiltonian bands with the characteristic spin-group integer and the even-parity wave form of altermagnetism, the crystallographic Laue group $\mathbf{G}$, the halving subgroup $\mathbf{H}$ of symmetry elements which interchange atoms between same-spin sublattices, a generator $A$ of symmetry elements which interchange atoms between opposite-spin sublattices, the nontrivial spin Laue group $R_s^{\mathrm{III}}$ (in brackets we list the number of symmetry elements), and material candidates of the third magnetic phase. The model Hamiltonian bands on which we illustrate the spin-momentum locking character are described in Supplemental Material Sec. III [30]. References describing the materials are in the main text and Supplemental Material Secs. V–VI [30].

Figure 3

Spin splitting by the electric crystal potential in ${\mathrm{KRu}}_4{\mathrm{O}}_8$ in the third magnetic phase. (a) Schematic spin arrangement on the ${\mathrm{KRu}}_4{\mathrm{O}}_8$ crystal with opposite-spin directions depicted by red and blue color. Magenta arrow and its label highlights the opposite-spin-sublattice transformation containing a real-space fourfold rotation. (b) Calculated spin-momentum locking with the characteristic spin-group integer 2 on top of two DFT Fermi surface sheets. (c),(d) DFT band structure of the nonmagnetic phase and the third magnetic phase, respectively. Gray shading highlights the $\mathbf{k}$-dependent splitting by the anisotropic electric crystal potential. (e),(f) Projection of bands on the sublattices $A$ and $B$ in the nonmagnetic phase (black) and third magnetic phase (red and blue) for the upper bands and lower bands, respectively. Color shading in (f) highlights the nearly $\mathbf{k}$-independent magnetic splitting of the lower bands, and its opposite sign for the sublattices $A$ and $B$ bands. (g) Real-space DFT spin density around the Ru atom in sublattices $A$ and $B$.

Figure 4

Metallic high critical temperature CrSb with the third magnetic phase. (a) Schematic crystal structure with DFT spin densities. Cr sublattices and the respective magnetization densities with opposite orientation of the magnetic moment are depicted by red and blue color. Magenta arrow and its label highlights the opposite-spin-sublattice transformation containing a real-space mirror or sixfold rotation. (b) Calculated bulklike spin-momentum locking with the characteristic spin-group integer 4 on top of two selected DFT Fermi surface sheets. (c) DFT band structure in the third magnetic phase. Wave-vector dependence of the spin splitting between the bands highlighted by the gray shading is plotted in the lower panel.