Perovskite Altermagnets: Symmetry & Spin Splitting

• Perovskite altermagnets are perovskite-structured materials exhibiting spin-split electronic bands from collinear antiferromagnetic order despite nearly zero net magnetization.
• They leverage crystal rotations, mirrors, and nonsymmorphic operations to lift spin Kramers degeneracy, resulting in anisotropic transport and magneto-optical responses.
• This family spans diverse systems such as Pnma perovskites, Ruddlesden–Popper phases, and 6H perovskites, enabling tunable electronic properties and coupled multipolar magnetic phenomena.

Perovskite altermagnets are perovskite-structure oxides, fluorides, Ruddlesden–Popper phases, and 6H perovskites in which collinear antiferromagnetic order yields non-relativistic spin-split electronic bands despite vanishing or nearly vanishing net magnetization. In the works surveyed here, the defining feature is that opposite-spin sublattices are connected by crystal rotations, mirrors, screws, or other unitary space operations rather than by pure translation or inversion combined with time reversal, so spin Kramers degeneracy is lifted over generic regions of the Brillouin zone. The resulting materials combine compensated magnetism with ferromagnetic-like band splitting, anisotropic transport, magneto-optical activity, and, in several cases, strong coupling to octahedral rotations, ferroelectric or antiferroelectric distortions, and higher-rank magnetic multipoles (Bernardini et al., 2024, Buiarelli et al., 27 Sep 2025, Streltsov et al., 31 Jul 2025).

1. Symmetry criteria and defining formulations

Within the cited literature, altermagnetism is formulated in two closely related ways. One formulation emphasizes a collinear antiferromagnet with staggered spin order of wave vector $\mathbf{q}=0$, zero net magnetization in the limit of vanishing spin–orbit coupling, and broken macroscopic time-reversal symmetry, so that the electronic bands exhibit non-relativistic spin splitting (Naka et al., 2024). A second formulation is cast in terms of spin-group or exchange-operation symmetry. Using the spin-resolved Kohn–Sham equation

$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$

an exchange operation $\mathbf{O}$ satisfies

$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$

If $\mathbf{O}k=k$, then $E_\uparrow(k)=E_\downarrow(k)$; if $\mathbf{O}k=k'\neq k$, then $E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$, which is the altermagnetic signature (Cui et al., 9 Jan 2026).

This distinction is important for perovskites because the relevant symmetry obstruction is not always encoded by a simple $\mathbf{q}=0$ statement. In cubic KMnF$_3$, for example, G-type antiferromagnetism is a zone-boundary order that preserves anti-translation, i.e. translation followed by time reversal, and therefore gives no spin splitting. In the tetragonal phase, the same-wavevector structural distortion $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$0 removes anti-translation symmetry when combined with the magnetic order parameter $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$1, converting the system into an altermagnet (Buiarelli et al., 27 Sep 2025).

Conventional collinear antiferromagnets instead preserve degeneracy-enforcing symmetries such as $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$2, inversion plus time reversal, or anti-translation. Ferromagnets break these symmetries and carry net magnetization. Perovskite altermagnets occupy the intermediate category: spin-split, zero-net-moment, and symmetry-compensated.

2. Structural families and representative material platforms

Recent work treats perovskite altermagnetism not as a single structural motif but as a family phenomenon spanning corner-sharing $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$3 perovskites, Ruddlesden–Popper derivatives, polar and antipolar perovskites, tetragonally distorted fluorides, and layered 6H hexagonal perovskites. In one broad survey of $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$4 compounds, high-throughput screening of the Materials Project database identified 140 candidate materials exhibiting antiferromagnetic behavior in the $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$5 space group; 91 preferentially stabilize in altermagnetic ground states, and 20 of these adopt the perovskite structure (Zhang et al., 9 Sep 2025). A complementary first-principles study showed that Ruddlesden–Popper and perovskite phases are generic hosts for altermagnetic behavior, including La$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$6NiO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$7, La$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$8Ni$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$9O$\mathbf{O}$0, BiFeO$\mathbf{O}$1, PbNiO$\mathbf{O}$2, BiNiO$\mathbf{O}$3, and mixed-anion nickelates (Bernardini et al., 2024). Additional work established antiferroelectric altermagnetism in BiCrO$\mathbf{O}$4, symmetry-driven multiferroic altermagnetism in Ca$\mathbf{O}$5Mn$\mathbf{O}$6O$\mathbf{O}$7, CaMnO$\mathbf{O}$8, and 2D Ca$\mathbf{O}$9Mn$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$0O$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$1, octahedral-rotation-induced altermagnetism in KMnF$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$2, and M-type and S-type altermagnetism in 6H perovskites A$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$3BB'$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$4O$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$5 (Duan et al., 2024, Cui et al., 9 Jan 2026, Buiarelli et al., 27 Sep 2025, Streltsov et al., 31 Jul 2025).

Platform	Representative compounds	Distinguishing feature
$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$6 perovskites	NaCoF$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$7, NaOsO$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$8, SrOsO$\mathbf{O}V_\uparrow \mathbf{O}^{-1}=V_\downarrow.$9, SrRhO$\mathbf{O}k=k$0, SrRuO$\mathbf{O}k=k$1	Nonsymmorphic spin-flipping operations and Berry-curvature responses
Ruddlesden–Popper / perovskite phases	La$\mathbf{O}k=k$2NiO$\mathbf{O}k=k$3, La$\mathbf{O}k=k$4Ni$\mathbf{O}k=k$5O$\mathbf{O}k=k$6, BiFeO$\mathbf{O}k=k$7, PbNiO$\mathbf{O}k=k$8	Octahedral tilts break $\mathbf{O}k=k$9 and $E_\uparrow(k)=E_\downarrow(k)$0
Fluoride perovskites	KMnF$E_\uparrow(k)=E_\downarrow(k)$1, RbMnF$E_\uparrow(k)=E_\downarrow(k)$2, K$E_\uparrow(k)=E_\downarrow(k)$3Rb$E_\uparrow(k)=E_\downarrow(k)$4MnF$E_\uparrow(k)=E_\downarrow(k)$5	Octahedral-rotation-induced altermagnetism
Multiferroic and antipolar perovskites	BiCrO$E_\uparrow(k)=E_\downarrow(k)$6, Ca$E_\uparrow(k)=E_\downarrow(k)$7Mn$E_\uparrow(k)=E_\downarrow(k)$8O$E_\uparrow(k)=E_\downarrow(k)$9, CaMnO$\mathbf{O}k=k'\neq k$0, Ca$\mathbf{O}k=k'\neq k$1Mn$\mathbf{O}k=k'\neq k$2O$\mathbf{O}k=k'\neq k$3	AFE/FE switching and multimode structural coupling
6H perovskites	Ba$\mathbf{O}k=k'\neq k$4CoIr$\mathbf{O}k=k'\neq k$5O$\mathbf{O}k=k'\neq k$6, Ba$\mathbf{O}k=k'\neq k$7SrIr$\mathbf{O}k=k'\neq k$8O$\mathbf{O}k=k'\neq k$9, Ba$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$0NiRu$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$1O$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$2, Ba$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$3TbRu$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$4O$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$5	Broken $E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$6 from local structural alternations in centrosymmetric lattices

Across these families, the recurrent crystallographic ingredients are octahedral tilts or rotations, layered alternation, mixed-anion or antipolar environments, and magnetic sublattices related by rotations or nonsymmorphic operations rather than by pure translation.

3. Structural induction and real-space multipoles

A central development in the perovskite literature is the shift from purely reciprocal-space descriptions of altermagnetism toward real-space magnetic multipoles. In KMnF$E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$7, high-temperature cubic $E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$8 symmetry changes near $E_\uparrow(k)\neq E_\downarrow(k)=E_\downarrow(k')$9 to tetragonal $\mathbf{q}=0$0, with MnF$\mathbf{q}=0$1 octahedra rotating around the $\mathbf{q}=0$2 axis by about $\mathbf{q}=0$3. The structural order parameter transforms as $\mathbf{q}=0$4, while the G-type magnetic order transforms as $\mathbf{q}=0$5. In the cubic phase, the magnetic order preserves anti-translation and there is no altermagnetic spin splitting. In the tetragonal phase, the same-wavevector structural distortion removes anti-translation when both $\mathbf{q}=0$6 and $\mathbf{q}=0$7 condense, and the altermagnetic splitting is proportional to both order parameters (Buiarelli et al., 27 Sep 2025).

The real-space signature of this transition is not a change in the integrated Mn spin dipole but a change in higher multipoles. The collinear spin density $\mathbf{q}=0$8 around Mn is aligned with the F octahedra and rotates with them, while the total integrated dipole per Mn, $\mathbf{q}=0$9, remains almost unchanged. By contrast, the lowest-order in-phase multipole that appears is the 32-pole

$_3$0

which grows linearly with the octahedral rotation angle. The work also identifies a symmetry-allowed trilinear coupling

$_3$1

and notes that the text also assigns the 32-pole to $_3$2, with the detailed label clarified in the Supplemental Material. The substantive point is that the induced higher-rank multipole is linear in the structural rotation amplitude and vanishes in the cubic limit (Buiarelli et al., 27 Sep 2025).

The same study generalizes this picture. Spin density around an atomic site is expanded in tesseral harmonics, with $_3$3 corresponding to the monopole $_3$4, $_3$5 to octupolar structures such as $_3$6, and $_3$7 to rank-5 32-poles such as $_3$8 or $_3$9. The decisive distinction is whether allowed multipoles are in-phase or out-of-phase between symmetry-related sites. Conventional antiferromagnets generally host out-of-phase higher multipoles that cancel over the unit cell; altermagnets host at least one in-phase higher-rank multipole that survives despite dipole cancellation. The same work also shows that even in nominally collinear magnets the local spin density is generically noncollinear when spin–orbit coupling is present: $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$00 and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$01 vanish only upon spatial integration, while noncollinear spin-density multipoles appear linearly in $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$02 (Buiarelli et al., 27 Sep 2025).

For $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$03 NaCoF$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$04, the real-space picture is complementary rather than identical. There, G-type antiferromagnetism coexists with G-type orbital order, and the symmetry relations between the Co sublattices are carried by operations such as $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$05 and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$06. The coupled orbital and magnetic texture underlies the d-wave-like altermagnetic Fermi-surface pattern seen after shifting $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$07 in calculations (Zhang et al., 9 Sep 2025).

4. Dimensionality, ferroelectricity, and antiferroelectricity

Perovskite altermagnetism is strongly dimensionality-dependent. In bulk Ca$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$08Mn$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$09O$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$10 with space group $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$11 and in bulk CaMnO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$12 with space group $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$13, A-, C-, and G-type antiferromagnetic orders all produce altermagnetism. In the 2D A$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$14B$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$15O$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$16 slab derived either from a layered Ruddlesden–Popper structure or by truncation of a GdFeO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$17-type perovskite, only C-type antiferromagnetism retains altermagnetic spin splitting. The decisive symmetry in the 2D geometry is $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$18: for A- and G-type AFM it remains an exchange operation mapping $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$19 and enforcing spin degeneracy, whereas for C-type AFM it does not directly connect opposite-spin sublattices at the same $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$20. In the fully relaxed 2D C-AFM phase of Ca$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$21Mn$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$22O$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$23, the altermagnetic spin splitting is on the order of $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$24 meV (Cui et al., 9 Jan 2026).

The same study shows that ferroelectric polarization and altermagnetic spin splitting are governed by the same structural modes. Mode decomposition around a high-symmetry $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$25 parent yields an in-plane polar mode $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$26, an in-phase octahedral rotation $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$27, an out-of-plane tilt $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$28, and a Jahn–Teller distortion $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$29. The polar mode is stable in isolation but becomes nonzero through multimode coupling with $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$30 and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$31, described by a trilinear free-energy term

$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$32

The altermagnetic splitting $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$33 is increased by $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$34, $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$35, and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$36 taken individually, while $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$37 alone does not generate spin splitting; yet the sum of the single-mode contributions is only $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$38 meV, much smaller than the $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$39 meV splitting of the fully relaxed ground state, so the full response is cooperative and multimode (Cui et al., 9 Jan 2026).

A parallel route to electrically controllable perovskite altermagnetism is antiferroelectricity. In BiCrO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$40, the orthorhombic $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$41 ground state hosts antipolar Bi displacements. In the antiferroelectric state, G-, A-, and C-type antiferromagnetic orders all become altermagnetic because the exchange operation connecting the spin sublattices is a screw $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$42, yielding a spin group of the form $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$43. In the ferroelectric state, Bi displacements align uniformly, the exchange operation reduces to pure translation $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$44, and the corresponding spin group is $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$45, which restores spin degeneracy at each $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$46. The cited work therefore defines BiCrO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$47 as an antiferroelectric altermagnet in its AFE phases and a conventional antiferromagnet in its FE phases, with electric-field switching expected to toggle altermagnetic spin splitting on and off (Duan et al., 2024).

Together these results define a distinct perovskite theme: ferroelectric, antiferroelectric, and rotational lattice modes do not merely coexist with magnetic order; they determine whether crystal symmetry allows or forbids altermagnetic splitting.

5. Electronic structure, transport, and optical response

The most systematic electronic-structure survey of $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$48 perovskites uses NaCoF$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$49 as a prototype. In this compound, G-type antiferromagnetism is the ground state over the tested $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$50–$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$51 eV range, with all antiferromagnetic states lower in energy than the ferromagnet. The non-relativistic band structure is spin-degenerate on the $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$52 and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$53 planes because of $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$54 and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$55, but it shows spin splitting at generic $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$56, including splittings of about 20 meV near $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$57 eV and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$58 eV in a $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$59 plane. After shifting the Fermi level to $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$60 eV in the calculation, the Fermi surface exhibits a d-wave pattern analogous to RuO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$61 (Zhang et al., 9 Sep 2025).

The same work connects these bands to response functions. In G-AFM NaCoF$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$62, symmetry leaves only $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$63 nonzero, so only the $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$64-component of Berry curvature survives, and the anomalous Hall, anomalous Nernst, and anomalous thermal Hall responses are correspondingly anisotropic. The optical conductivity is likewise anisotropic: $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$65 has peaks $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$66 at $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$67 eV and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$68 at $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$69 eV, while $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$70 has $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$71 around 3.5 eV and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$72 around 4.3 eV. For G-AFM NaCoF$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$73, the Kerr rotation reaches $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$74 degrees, and the Faraday rotation per unit length reaches $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$75 degrees/cm (Zhang et al., 9 Sep 2025).

Ruddlesden–Popper and perovskite nickelates and ferrites supply larger non-relativistic splittings and a different perspective on altermagnetic k-space topology. In La$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$76NiO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$77, the maximal valence-band splitting is about 161 meV. In metallic La$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$78NiO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$79F$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$80, the splittings are 86 meV at $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$81, 112 meV for bands crossing $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$82, and 194 meV in the valence bands. In metallic La$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$83Ni$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$84O$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$85, the corresponding values are 28, 48, and 121 meV. PbNiO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$86 reaches 334 meV in the valence bands, and BiFeO$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$87 reaches 316 meV. These calculations also reveal accidental nodes and distinct topologies in the spin-momentum texture at the Brillouin-zone boundary, which the authors present as a refinement beyond the usual d-wave or higher even-parity-wave labels (Bernardini et al., 2024).

The 6H perovskites extend the response taxonomy. In Ba$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$88CoIr$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$89O$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$90, DFT+U without SOC gives a metallic altermagnet with spin splitting along $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$91, while DFT+U+SOC opens a gap of about 40 meV and induces a net moment of $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$92 per formula unit, approximately along $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$93, with canting angles larger than $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$94 on Co and $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$95 on Ir. Its MPG $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$96 allows a single off-diagonal optical conductivity component, $$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$97. In Ba$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$98NiRu$$\left[\frac{1}{2}(k-i\nabla)^2+V_\sigma\right]\psi_\sigma(k)=E_\sigma(k)\psi_\sigma(k),$$99O$\mathbf{O}$00, the S-type MPG $\mathbf{O}$01 forbids Hall and magneto-optical responses but allows piezomagnetism. Under 1% tensile strain along $\mathbf{O}$02 and hole doping, the calculated magnetization remains purely along $\mathbf{O}$03; at $\mathbf{O}$04 per unit cell, the Ru contribution reaches about $\mathbf{O}$05 per Ru, for roughly $\mathbf{O}$06 per unit cell in total, with a small compensating Ni contribution of $\mathbf{O}$07 (Streltsov et al., 31 Jul 2025).

6. Tunability, classification issues, and open directions

A distinctive feature of perovskite altermagnets is that their phase space can be navigated structurally, chemically, and dimensionally. KMnF$\mathbf{O}$08 and RbMnF$\mathbf{O}$09 illustrate this directly: RbMnF$\mathbf{O}$10 has a Néel temperature comparable to that of KMnF$\mathbf{O}$11 but remains cubic at all temperatures, so K$\mathbf{O}$12Rb$\mathbf{O}$13MnF$\mathbf{O}$14 is proposed as a model system in which the structural transition is lowered and separated from the magnetic one, enabling controlled study of an antiferromagnetic-to-altermagnetic transition (Buiarelli et al., 27 Sep 2025). In 2D perovskites, when the magnetic ground state is not C-type, the cited design rules to recover altermagnetism are to break $\mathbf{O}$15, $\mathbf{O}$16, $\mathbf{O}$17, or $\mathbf{O}$18 before imposing magnetic order, or to preserve a symmetry that cannot act as an exchange operation after the magnetic pattern is fixed. The explicit strategies proposed are superlattice engineering, shear strain, electric field, and substrate engineering (Cui et al., 9 Jan 2026).

The literature also raises classification questions. One study argues that the traditional ferro-, ferri-, and antiferromagnet taxonomy is formally incomplete once non-relativistic spin splitting is admitted in zero- or near-zero-moment states. BiNiO$\mathbf{O}$19, for example, is described as a Luttinger-compensated ferrimagnet whose ferromagnetic-like properties are attributed primarily to the antiferromagnetic component via altermagnetism. BiFeO$\mathbf{O}$20 is presented as a “failed altermagnet” in bulk because its long-wavelength spiral washes out the underlying $\mathbf{O}$21-type altermagnetic splitting, while thin films can truncate the spiral and restore the altermagnetic state (Bernardini et al., 2024).

Several limitations recur across the cited studies. The real-space multipole analysis emphasizes atomic multipoles inside atomic spheres because unit-cell multipoles in periodic solids do not have well-established gauge-invariant formulations. Most results are obtained at the DFT or DFT+U+SOC level, often with static structural and magnetic order, and finite-temperature effects are typically treated only indirectly through known transition temperatures or phenomenological phase arguments (Buiarelli et al., 27 Sep 2025). This suggests that the central open problems are not definitional but quantitative: how robust the predicted spin splitting and response tensors are to fluctuations, how reliably structural tuning can switch between AFM and altermagnetic phases, and how these states couple to superconductivity, multiferroicity, and surface-sensitive probes.

What is already established is more specific than a general promise. Perovskite altermagnets now encompass $\mathbf{O}$22 perovskites with symmetry-protected Berry-curvature responses, RP nickelates with sizable non-relativistic spin splitting and nontrivial Brillouin-zone-boundary textures, multiferroic and antiferroelectric systems in which electric order toggles altermagnetic band splitting, fluorides in which octahedral rotations induce 32-polar order, and 6H perovskites in which centrosymmetric lattices host either magneto-optical M-type or giant-piezomagnetic S-type altermagnetism. Single crystals of 6H perovskites are described as readily grown and cleavable, and the layered and thin-film perovskite platforms provide direct routes to spin-resolved STM, spin-resolved ARPES, strain control, and heterostructure engineering (Streltsov et al., 31 Jul 2025).