M-Type Altermagnet: Orbital Ferrimagnets

• M-type altermagnets are magnetic states with fully compensated spin moments that acquire a net orbital moment from spin–orbit coupling under broken T and PT symmetries.
• They display ferromagnetic point group characteristics and linear anomalous Hall effects, reconciling antiferromagnetic spin cancellation with ferromagnetic transport signatures.
• Model systems such as ilmenite CoMnO₃ illustrate strong non-relativistic spin-split bands in M-type altermagnets, advancing our understanding of orbital ferrimagnetism.

An M-type altermagnet is the moment-bearing branch of a symmetry-centered altermagnet classification in which altermagnets are defined as magnetic states with fully compensated spin angular momenta and broken $PT$ symmetry. In that scheme, M-type altermagnets have broken $T$ symmetry and nonzero net magnetic moments, yet their spin moments remain compensated; the net moment is attributed to orbital angular momenta originating from SOC, so M-type altermagnets are described as orbital ferrimagnets and lie within the ferromagnetic point group (Cheong et al., 2024, Cheong et al., 20 Mar 2025). The concept is therefore distinctive in that it combines an altermagnetic, spin-compensated background with ferromagnetic-point-group behavior and, in strong cases, non-relativistic spin-split bands.

1. Definition and conceptual scope

The symmetry-centered classification that explicitly uses the label M-type broadens altermagnetism beyond the narrow identification with compensated, zero-moment antiferromagnet-like states. In that formulation, the decisive condition is broken $PT$ together with fully compensated spins, rather than vanishing total magnetic moment in every case (Cheong et al., 2024). M-type altermagnets then occupy the subclass in which $T$ is broken and a net magnetic moment is present.

This produces an apparent tension: if spins are fully compensated, how can the material carry a net moment? The resolution given in the classification literature is that the moment is not an uncompensated spin moment. Instead, the ferromagnetic behavior is said to arise solely from orbital angular momenta due to SOC, which is why M-type altermagnets are characterized as orbital ferrimagnets (Cheong et al., 2024, Cheong et al., 20 Mar 2025). In this sense, M-type altermagnetism is distinct from conventional ferromagnetism, conventional ferrimagnetism, and conventional antiferromagnetism at once.

A related literature on extended altermagnetism organizes broken-$PT$ compensated magnets into type-I, type-II, and type-III classes rather than M/S/A labels. Within the supplied data, the moment-bearing branch is the nearest analogue of type-I, which is explicitly described as belonging to the ferromagnetic point group and as a form of weak ferromagnetism (Cheong et al., 2024, Cheong et al., 2024). This suggests that “M-type” and “type-I” are closely aligned usages in the moment-bearing sector, although the papers do not present a single universal taxonomy.

2. Symmetry criteria and classification structure

The explicit three-way classification is given in a compact form as follows (Cheong et al., 2024):

Class	Symmetry statement	Net magnetic moment
M-type	broken $T$ symmetry	nonzero
S-type	broken $T$ symmetry	zero
A-type	unbroken $T$ and broken $P$ symmetries	not specified in the class label

For M-type, the formal identification rule in the classification details is: broken $PT$, broken $T$0, membership in the ferromagnetic point group, and fully compensated spin angular momenta (Cheong et al., 2024). The same source states that 31 magnetic point groups belonging to the ferromagnetic point group can be Type-I altermagnets if they have fully compensated spins, giving the magnetic-point-group criterion for the moment-bearing branch (Cheong et al., 2024).

This framework differs from the more common material-specific language that defines altermagnets as compensated magnets with vanishing net magnetization. In the M-type formulation, the zero-net-spin condition remains essential, but the total moment need not vanish because orbital magnetization can survive once SOC is included (Cheong et al., 2024, Cheong et al., 20 Mar 2025). A plausible implication is that M-type altermagnetism is best understood as a broken-$T$1, spin-compensated, orbital-moment-bearing sector rather than as a simple variant of compensated antiferromagnetism.

The non-collinear extension of altermagnetism is also relevant here. The extended framework argues that altermagnetism does not have to be limited to 2 alternating directors and collinear antiferromagnetic spins, but can include multiple directors and non-collinear spins so long as the pure spin moments remain fully compensated in the zero-SOC limit (Cheong et al., 2024). This allows moment-bearing, ferromagnetic-point-group altermagnets to appear in geometries beyond the canonical two-sublattice collinear case.

3. Strong and weak M-type altermagnets

A second classification layer distinguishes strong from weak altermagnets. In the symmetry-centered treatment, strong altermagnets have spin-split bands through exchange coupling in the non-relativistic limit, i.e. for zero SOC, whereas weak altermagnets have spin-split bands only with non-zero SOC (Cheong et al., 2024). The criterion is given in terms of the number of symmetric orthogonal spin-rotation operations $T$2, $T$3, and $T$4: a system cannot have spin-split bands for zero SOC if it has two or more such spin-rotation symmetries (Cheong et al., 2024).

For the moment-bearing branch, the key result is that collinear Type-I/M-type altermagnets are always strong (Cheong et al., 2024). The reasoning supplied is that for collinear spins, only one orthogonal spin-rotation symmetry remains unbroken, which is sufficient to allow non-relativistic spin splitting. The same source adds that the same conclusion likely extends to non-collinear type-I systems, although the exact proof depends on the magnetic unit cell (Cheong et al., 2024).

This distinction matters because two different relativistic roles coexist in M-type altermagnetism. First, the net magnetic moment is described as orbital/SOC-derived. Second, in the strong case, the spin-split bands can nevertheless persist already in the zero-SOC limit (Cheong et al., 2024). That coexistence is one of the defining conceptual features of the class.

4. Electronic structure and response phenomenology

The electronic hallmark of M-type altermagnets is the coexistence of spin compensation with spin-split bands under broken $T$5 symmetry (Cheong et al., 2024, Cheong et al., 20 Mar 2025). In the SAM classification for kinetomagnetism and altermagnetism, M-type altermagnets are described as having broken $T$6, belonging to the ferromagnetic point group, and exhibiting orbital ferrimagnetism with uncompensated magnetization (Cheong et al., 20 Mar 2025). The same literature emphasizes that the spin sector remains fully compensated.

The most characteristic transport consequence is that all M-type altermagnets show linear AHE (Cheong et al., 2024). In the current-induced-magnetization language, M-type altermagnets also exhibit longitudinal even-order current-induced magnetization and transverse even-order current-induced magnetization, while some M-type states with broken $T$7, $T$8, and $T$9 can additionally support transverse odd-order current-induced magnetization and even-order AHE (Cheong et al., 20 Mar 2025). This places M-type altermagnets at the ferromagnetic-like end of the altermagnetic response spectrum.

A central conceptual point is that this response phenomenology is not attributed to uncompensated spin ferromagnetism. The classification literature instead argues that M-type altermagnets behave ferromagnetically at the macroscopic symmetry level because SOC generates uncompensated orbital angular momentum, while the spin angular momenta remain compensated (Cheong et al., 2024, Cheong et al., 20 Mar 2025). This makes them distinct from ordinary ferromagnets, in which the net moment is carried directly by spin polarization, and from conventional ferrimagnets, in which unequal sublattice spins produce the moment.

The non-collinear extension of altermagnetism further enlarges the possible response landscape. Type-I moment-bearing states in non-collinear settings are reported with allowed linear AHE, Faraday rotation, and MOKE, again reflecting ferromagnetic-point-group symmetry rather than uncompensated spin magnetism in the ordinary sense (Cheong et al., 2024).

5. Model systems and explicit examples

Several explicit examples of moment-bearing altermagnets or closely allied type-I states appear in the supplied literature. The clearest material example is ilmenite CoMnO$PT$0, identified as an M-type altermagnet with $PT$1, large magnetic anisotropy, and orbital ferrimagnetism, while the spin angular momenta of Mn$PT$2 and Co$PT$3 are canceled (Cheong et al., 20 Mar 2025). This example directly realizes the defining M-type combination of spin compensation and nonzero net magnetization.

The same response-classification literature also identifies non-collinear kagome examples corresponding to the moment-bearing branch: a state with MPG $PT$4 realized in Mn$PT$5Ge(Ga) and a state with MPG $PT$6 realized in Mn$PT$7Sn (Cheong et al., 2024). These are important because they show that the moment-bearing altermagnetic sector is not restricted to collinear two-sublattice motifs.

A further explicit example is the supplementary $PT$8 case realizable in Mn$PT$9Mo$T$0O$T$1, described as a type-I altermagnet with broken $T$2, $T$3, and $T$4, a net electric polarization along $T$5, a net magnetic moment along $T$6, linear AHE, transverse odd-order current-induced magnetization, and even-order AHE (Cheong et al., 2024). This makes it a particularly rich realization of moment-bearing altermagnetism embedded in a polar lattice environment.

The model side is equally important. The classification literature discusses explicit type-I model structures on square and kagome lattices, including collinear and non-collinear patterns with magnetic point groups such as $T$7, $T$8, $T$9, and $PT$0 (Cheong et al., 2024, Cheong et al., 2024). These examples are used to demonstrate that the M-type/type-I branch can arise from either $PT$1-tensor anisotropy or DM interaction, provided the pure spin moments are fully compensated in the absence of SOC (Cheong et al., 2024).

6. Terminology in the wider altermagnet literature

A persistent source of confusion is that many recent material-specific altermagnet papers do not use the term M-type at all. Instead, they classify systems by the angular character of spin symmetry—$PT$2-, $PT$3-, or $PT$4-wave altermagnetism—or by spin Laue groups, magnetic space groups, or other symmetry constructions. For MnTe, for example, a 2025 spin-ARPES study explicitly states that it does not introduce an “M-type altermagnet” category and instead treats MnTe as a $PT$5-wave altermagnet (Din et al., 3 Nov 2025). Likewise, the perovskite review discusses $PT$6-wave altermagnets rather than M-type labels (Naka et al., 2024), and the hematite MOKE paper classifies $PT$7-Fe$PT$8O$PT$9 as a $T$0-wave altermagnet candidate with spin Laue group $T$1, again without any M-type terminology (Luo et al., 10 Dec 2025).

This terminological divergence is not merely stylistic. In the M-type/S-type/A-type scheme, the classification variable is the relation between $T$2, $T$3, and net magnetic moment (Cheong et al., 2024). In the $T$4-wave literature, the classification variable is the angular character of the momentum-dependent spin splitting or nodal structure (Din et al., 3 Nov 2025, Naka et al., 2024). These are different symmetry cuts through the same broader field. A plausible implication is that “M-type” and “$T$5-wave,” for example, are not competing names for the same object, but labels drawn from different classification layers.

Recent extensions of altermagnetism deepen this diversity. “Atomic altermagnetism” introduces even-parity, ferroically ordered, non-dipolar spin density on atomic sites and does not use M-type language (Jaeschke-Ubiergo et al., 13 Mar 2025). “Spin-orbital altermagnetism” instead introduces intrinsic and extrinsic spin-orbital subclasses, again without any M-type taxonomy (Wang et al., 19 Sep 2025). Accordingly, material papers that discuss MnTe, hematite, perovskites, or atomic and spin-orbital forms of altermagnetism should not be retroactively cited as evidence for an M-type designation unless the source itself adopts that scheme.

The safest encyclopedic conclusion is therefore twofold. First, M-type altermagnet is a legitimate, explicit term in a symmetry-centered branch of the recent literature, where it denotes a broken-$T$6, broken-$T$7, moment-bearing, spin-compensated altermagnet with ferromagnetic-point-group character (Cheong et al., 2024, Cheong et al., 20 Mar 2025). Second, this usage coexists with a larger body of altermagnet research that classifies materials by $T$8-wave symmetry, spin Laue groups, type-I/II/III, or newer constructs such as atomic and spin-orbital altermagnetism, often without any M-type label at all (Cheong et al., 2024, Din et al., 3 Nov 2025).