Orbital Altermagnetic Phases

• Orbital altermagnetic phases are magnetic states characterized by orbital ordering that induces momentum-dependent band anisotropy and compensates magnetic moments in real space.
• They arise from mechanisms such as spontaneous orbital order and staggered loop currents, leading to distinct d-wave or g-wave spin splitting in various lattice setups.
• Experimental and theoretical approaches, including spin-resolved ARPES and model Hamiltonians, are used to investigate their topological, superconducting, and fractionalized extensions.

Orbital altermagnetic phases are magnetic states in which orbital degrees of freedom are the symmetry source, the microscopic order parameter, or the experimentally visible channel of altermagnetism. In current usage, the term encompasses at least three closely related situations: collinear antiferromagnets whose nonrelativistic spin splitting is generated by spontaneous orbital ordering rather than by pre-existing crystallographic sublattice inequivalence; pure orbital magnetic orders built from staggered loop currents or sublattice-resolved orbital magnetization; and fractionalized phases in which spin-rotation-invariant orbital observables retain altermagnetic symmetry even after long-range spin order is lost (Meier et al., 3 Feb 2025, Pan et al., 1 Oct 2025, Sobral et al., 2024). Across these settings, the defining motif is compensation in real space together with momentum-dependent band or response anisotropy that is fixed by point-group symmetry rather than by uniform ferromagnetic polarization.

1. Definitions and conceptual scope

Altermagnetism is a collinear magnetic order with zero net magnetization but nonrelativistic spin-split bands. This distinguishes it from conventional collinear antiferromagnets, where translation- or inversion-related sublattices typically enforce spin degeneracy, and from ferromagnets, where spin splitting is tied to a net moment. Orbital altermagnetic phases extend this logic to cases where orbital structure either generates the spin splitting or becomes the primary ordered quantity itself (Meier et al., 3 Feb 2025, Leeb et al., 2023).

A first meaning of orbital altermagnetism is orbital-order-driven spin altermagnetism. In this case, spontaneous orbital order breaks the sublattice equivalence electronically, so that a high-symmetry lattice can nevertheless host altermagnetic spin splitting without spin-orbit coupling. The Ruddlesden–Popper chromates provide the clearest formulation: orbital order in the $d_{xz}/d_{yz}$ sector lifts the effective sublattice equivalence of a collinear antiferromagnet and thereby generates a layer-resolved d-wave-like spin splitting (Meier et al., 3 Feb 2025).

A second meaning is pure orbital altermagnetism. Here the ordered moments are orbital magnetic moments rather than spin moments. Real-space compensation takes the form of anti-parallel orbital moments on sites or plaquettes, while momentum space exhibits orbital band splitting or orbital-momentum locking. In the square-kagome model this is realized by staggered loop currents, whereas in the honeycomb setting it is diagnosed by sublattice-resolved orbital magnetizations $M_A$ and $M_B$, with $M_z=M_A+M_B$ and $M_z^s=M_A-M_B$; the orbital-antiferromagnetic or “orbital altermagnetic” case is $M_z=0$ with $M_z^s\neq 0$ (Pan et al., 1 Oct 2025, Lage et al., 11 Jul 2026).

A third meaning appears in fractionalized systems. There, time-reversal breaking survives in spin-rotation-invariant observables such as staggered scalar chirality or orbital loop currents, while total magnetization still cancels by point-group symmetry. This is the sense in which “orbital altermagnetic spin liquid” and “altermagnetic spin liquid” are used in frustrated-magnet models and in exactly solvable Majorana constructions (Sobral et al., 2024, Sobral et al., 30 Dec 2025).

A further refinement is anti-altermagnetism. In the chromate literature, anti-altermagnetism denotes a compensated state in which each layer is locally altermagnetic, but adjacent layers carry opposite spin splitting so that the bulk splitting vanishes. Local altermagnetic behavior persists, yet the macroscopic band structure is spin-degenerate (Meier et al., 3 Feb 2025).

2. Microscopic mechanisms

The most explicit microscopic route is the coexistence of staggered spin and orbital order in multiorbital lattices. In the chromates, the relevant layer-resolved order parameters are

$$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$

with $\Delta n=n_{xz}-n_{yz}$. The altermagnetic character is then classified by

$$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$

If $M_A$0, the bulk exhibits net spin splitting; if $M_A$1 while $M_A$2 layerwise, the system is anti-altermagnetic (Meier et al., 3 Feb 2025). In the same spirit, a two-orbital square-lattice model with $M_A$3 orbitals shows that pure antiferromagnetism or pure orbital order leaves the bands spin-degenerate, whereas their coexistence produces an even-parity d-wave spin splitting (Leeb et al., 2023).

A complementary wavefunction-level viewpoint emphasizes orbital anisotropy in same-spin hopping channels. In orbital-engineered square lattices, single-orbital bipartite antiferromagnets remain spin-degenerate because antiunitary symmetries still relate the two spin sectors, but interwoven dual-orbital patterns lift this degeneracy. Interwoven $M_A$4 configurations generate d-wave splitting, while opposite-chirality mixtures of $M_A$5 and $M_A$6 generate g-wave splitting. The low-energy form factors are

$M_A$7

and

$M_A$8

near $M_A$9 (Che et al., 24 May 2026).

Pure orbital altermagnetism is generated differently. In the square-kagome model, complex hoppings induce staggered loop currents whose bond-current expectation values

$M_B$0

circulate with opposite chirality on neighboring plaquettes. The resulting orbital splitting obeys

$M_B$1

so the orbital sector itself acquires a d-wave-like momentum locking (Pan et al., 1 Oct 2025). In amorphous systems, the same basic outcome does not require a global crystal rotation: a common local point-group rotation acting on directional orbitals is sufficient. With two local orbitals $M_B$2, the collinear spin-orbital order parameter

$M_B$3

is invariant under the combined local orbital rotation and spin flip, even though neither operation is preserved separately (d'Ornellas et al., 11 Apr 2025).

3. Symmetry classes and momentum-space structures

The symmetry content of orbital altermagnetic phases is broader than the original even-parity d-wave paradigm. In two-dimensional pure orbital altermagnets with in-plane ferromagnetic spins, $M_B$4 and $M_B$5 must be absent because either symmetry would force $M_B$6. The magnetic point groups explicitly identified as compatible with orbital altermagnetism and ferromagnetism are $M_B$7, $M_B$8, $M_B$9, $M_z=M_A+M_B$0, $M_z=M_A+M_B$1, and $M_z=M_A+M_B$2 (Pan et al., 1 Oct 2025).

Within square-lattice settings, the central distinction is between d-wave and g-wave even-parity splitting. The d-wave form $M_z=M_A+M_B$3 vanishes on $M_z=M_A+M_B$4, while the g-wave form $M_z=M_A+M_B$5 or its lattice analogue vanishes on all high-symmetry lines and changes sign in eight sectors of the Brillouin zone (Che et al., 24 May 2026). The chromate and square-lattice two-orbital mechanisms both realize the d-wave case, whereas interwoven rotated $M_z=M_A+M_B$6-orbital patterns in square monolayers and metal-organic frameworks realize the g-wave case (Meier et al., 3 Feb 2025, Che et al., 24 May 2026).

Odd-parity altermagnetism is a separate symmetry class. In stacked noncentrosymmetric bilayers with interlayer antiferromagnetism and an in-plane layer flip, the band energies satisfy

$M_z=M_A+M_B$7

while generically $M_z=M_A+M_B$8. The protecting antiunitary symmetry is

$M_z=M_A+M_B$9

which acts as an effective time-reversal symmetry despite explicit breaking of ordinary $M_z^s=M_A-M_B$0. This construction supports both $M_z^s=M_A-M_B$1- and $M_z^s=M_A-M_B$2-wave odd-parity altermagnets (Zhuang et al., 25 Aug 2025).

The same symmetry logic also generalizes compensation beyond two-sublattice settings. On the kagome lattice, an odd number of crystallographic sublattices does not preclude altermagnetic-like compensation if the orbital moments are non-uniform within an enlarged charge-ordered cell. In that case, mirror-related plaquettes carry opposite orbital moments and the spin splitting inherits a d-wave nodal structure once spin-orbit coupling transfers orbital symmetry into the spin sector (Chakraborty et al., 30 Sep 2025).

4. Material platforms and concrete realizations

The Ruddlesden–Popper chromates $M_z^s=M_A-M_B$3, including $M_z^s=M_A-M_B$4, constitute the canonical orbital-order-driven platform. Their formal $M_z^s=M_A-M_B$5 $M_z^s=M_A-M_B$6 configuration favors $M_z^s=M_A-M_B$7, and spontaneous alternating $M_z^s=M_A-M_B$8 occupation couples to collinear antiferromagnetism. Odd-$M_z^s=M_A-M_B$9 members ($M_z=0$0) show clear nonrelativistic spin splitting, whereas even-$M_z=0$1 members ($M_z=0$2) and the perovskite limit remain globally compensated and thus anti-altermagnetic. Increasing $M_z=0$3 favors metallicity, and in $M_z=0$4 the altermagnetic and anti-altermagnetic configurations are nearly degenerate, with a computed energy difference of $M_z=0$5 (Meier et al., 3 Feb 2025).

A spectroscopic interfacial realization was proposed for the $M_z=0$6 heterostructure. Resonant inelastic x-ray scattering resolves an interfacial magnon mode $M_z=0$7 with maximum energy $M_z=0$8, weak dispersion, and intensity that rises sharply as $M_z=0$9, unlike a plain antiferromagnet. The data are consistent with checkerboard orbital order in the interfacial $M_z^s\neq 0$0 layer, correlated with the Néel sublattices. The extracted in-plane exchange is strongly reduced at the interface, $M_z^s\neq 0$1, compared with $M_z^s\neq 0$2 in bulk-like layers (Sarkar et al., 2024).

Pure orbital altermagnetism has been identified in first-principles studies of monolayer $M_z^s\neq 0$3, $M_z^s\neq 0$4, $M_z^s\neq 0$5, and $M_z^s\neq 0$6. In $M_z^s\neq 0$7, opposite-sign $M_z^s\neq 0$8 resides on two Cu sites even though the spin order is uniformly in-plane ferromagnetic; along $M_z^s\neq 0$9, the computed $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$0 reaches about $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$1. In $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$2, symmetry forbids on-site $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$3, but inter-site currents of $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$4 and $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$5 circulate on V–S–V–S rhombi, generating a loop-type orbital altermagnet. In $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$6, the nonlinear orbital response shows a pronounced peak at $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$7 (Pan et al., 1 Oct 2025).

Several lattice-model platforms broaden the scope further. In the modified Haldane model on the honeycomb lattice, the $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$8 state is a $$L_i=S_A^i-S_B^i,\qquad \Lambda_i=\Delta n_A^i-\Delta n_B^i,$$9-symmetric orbital antiferromagnet with $\Delta n=n_{xz}-n_{yz}$0 and $\Delta n=n_{xz}-n_{yz}$1, while tuning $\Delta n=n_{xz}-n_{yz}$2 interchanges orbital ferro- and antiferromagnetic character in insulating regimes (Lage et al., 11 Jul 2026). In kagome metals relevant to the $\Delta n=n_{xz}-n_{yz}$3 family, intertwined triple-$\Delta n=n_{xz}-n_{yz}$4 charge-density-wave and loop-current order can stabilize orbital ferromagnetic, antiferromagnetic, and pure d-wave altermagnetic states inside the $\Delta n=n_{xz}-n_{yz}$5 charge-ordered phase (Chakraborty et al., 30 Sep 2025). In $\Delta n=n_{xz}-n_{yz}$6 metal-organic framework monolayers $\Delta n=n_{xz}-n_{yz}$7-TCNX, especially flat Fe–TCNE, first-principles calculations show g-wave altermagnetism with a maximum spin splitting of about $\Delta n=n_{xz}-n_{yz}$8 along a generic momentum path while high-symmetry lines remain degenerate (Che et al., 24 May 2026).

5. Topological orbital altermagnetic phases

Topological variants arise when orbital altermagnetic textures are combined with band inversion, inversion breaking, or effective time-reversal symmetries. A ferroelectric altermagnetic Chern-insulator construction starts from a BHZ-type model and adds a d-wave altermagnetic term, spin canting, ferroelectric orbital hybridization, and Rashba spin-orbit coupling. In the pure altermagnetic limit, the d-wave form factor vanishes at $\Delta n=n_{xz}-n_{yz}$9, so the two spin sectors remain degenerate there. Spin canting lifts this degeneracy, and the resulting odd-Chern window is controlled by

$$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$0

Ferroelectric polarization further broadens the $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$1 region, and the full phase diagram contains $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$2 and $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$3 phases with electrically controlled Berry-curvature reorganization and orbital magnetization (Tagani et al., 10 Jun 2026).

Odd-parity orbital altermagnets provide a second topological route. In bilayers made from noncentrosymmetric monolayers with interlayer antiferromagnetism, orbital-phase hopping generates momentum-odd spin splitting without conventional SOC. On square lattices this yields $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$4-wave altermagnets; on hexagonal lattices it yields $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$5-wave altermagnets with

$$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$6

near $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$7. In spin-conserving regimes the corresponding quantum spin Hall phases carry helical edge states and quantized spin Hall conductance

$$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$8

These phases are protected by the effective antiunitary symmetry $$\boldsymbol{\Lambda}\cdot\boldsymbol{L}=\sum_i \Lambda_i L_i.$$9, not by ordinary time reversal (Zhuang et al., 25 Aug 2025).

The honeycomb orbital-antiferromagnetic setting is topological in a different sense. There, the standard and modified Haldane models realize distinct valley mechanisms: valley-dependent Dirac masses in the standard model and valley-dependent energy shifts in the modified model. This distinction explains why the modified model yields a $M_A$00-symmetric orbital antiferromagnet at $M_A$01, while the standard model yields an orbital ferromagnet under the same condition (Lage et al., 11 Jul 2026).

6. Superconducting and fractionalized extensions

Orbital altermagnetism also appears in superconducting settings. In a square-lattice unconventional superconductor with competing $M_A$02-wave and extended $M_A$03-wave pairing, a square adatom superlattice can stabilize a time-reversal-breaking superconducting state with loop currents that preserve superlattice translations but are odd under $M_A$04. The bond-current operator is

$M_A$05

and the typical maximal current in the orbital-altermagnetic superconducting state is $M_A$06. The resulting Bogoliubov bands have a finite Berry-curvature quadrupole, and with Rashba SOC they acquire altermagnetic spin splitting and nontrivial spin textures while keeping zero net spin moment (Pupim et al., 2024).

Fractionalized orbital altermagnetism was formulated on the checkerboard lattice in terms of a noncoplanar phase with staggered scalar chirality. In that phase, spin-rotation-invariant observables are isotropic in nearest-neighbor spin products but have

$M_A$07

and preserve $M_A$08 and $M_A$09. When quantum fluctuations melt the ordered phase, the proximate $M_A$10 spin liquid restores global $M_A$11 while retaining the same altermagnetic chirality pattern. Upon doping, the electronic spectral function shows split Fermi surfaces even though the electron Green function is diagonal in spin and the total spin polarization vanishes, a phenomenon described as “fractionalized spin-orbit coupling” (Sobral et al., 2024).

Exactly solvable Majorana models sharpen this fractionalized picture. On the square-octagon lattice, a spin-$M_A$12 model realizes a unique g-wave orbital altermagnetic spin liquid; on the checkerboard lattice, a spin-$M_A$13 model supports a d-wave orbital altermagnetic spin liquid competing with chiral and stripy phases. In both cases time reversal is broken by flux patterns and circulating currents, local plaquette orbital moments are nonzero, but the net orbital moment vanishes by point-group symmetry (Sobral et al., 30 Dec 2025).

A related superconducting extension comes from altermagnetic fluctuations in inversion-broken two-orbital systems. There, AM fluctuations favor a spin-triplet state with an internal $M_A$14-phase difference,

$M_A$15

equivalently a $M_A$16-type order parameter $M_A$17. The paper interprets this as a general inter-orbital spin-triplet superconductivity tied to the orbital structure of the altermagnetic fluctuations (Lu et al., 21 Oct 2025).

7. Experimental diagnostics, tunability, and open questions

The experimental toolkit depends on whether the primary order is spin-orbital, purely orbital, or fractionalized. For orbital-order-driven altermagnets, spin-resolved ARPES is the direct probe of nonrelativistic band splitting without net magnetization; in odd-$M_A$18 chromates it should detect spin splitting, whereas anti-altermagnetic members should look globally compensated unless layer-selective methods such as standing-wave ARPES resolve the layer-local splitting (Meier et al., 3 Feb 2025). At oxide interfaces, polarization-resolved RIXS is already sensitive to the anomalous magnon form factors produced by intertwined spin and orbital order (Sarkar et al., 2024). For pure orbital altermagnets, Kerr and Faraday effects, XMCD, resonant x-ray scattering, and magneto-optical probes target orbital moments and loop currents, while scanning SQUID or NV magnetometry can image the real-space $M_A$19 pattern generated by superconducting or loop-current states (Pan et al., 1 Oct 2025, Pupim et al., 2024).

Tunability is unusually rich because the order is tied to orbital character. In $M_A$20, orthorhombic epitaxial strain $M_A$21 couples as

$M_A$22

with $M_A$23 and $M_A$24, and therefore favors $M_A$25, i.e. the altermagnetic phase. More generally, layer parity, superlattice engineering, dimensional reduction, ferroelectric polarization, and ligand-controlled orbital rotation all serve as control parameters across chromates, ferroelectric Chern models, and metal-organic frameworks (Meier et al., 3 Feb 2025, Tagani et al., 10 Jun 2026, Che et al., 24 May 2026).

Several central questions remain open. In the chromates, quantitative predictions depend on $M_A$26 and $M_A$27, explicit meV-scale spin splittings were not tabulated, and the small $M_A$28 energy difference in $M_A$29 implies competing domains at finite temperature (Meier et al., 3 Feb 2025). In the honeycomb orbital-antiferromagnetic setting, directly measuring $M_A$30 and $M_A$31 separately is nontrivial, even though the real-space formulation defines them cleanly (Lage et al., 11 Jul 2026). In amorphous systems, finite-temperature behavior and microscopic derivations of the effective spin-orbital coupling $M_A$32 remain open (d'Ornellas et al., 11 Apr 2025). In fractionalized phases, spectral weights, lifetimes, and gauge-fluctuation effects beyond saddle point remain unresolved, as does the direct detection of visons and other topological excitations (Sobral et al., 2024, Sobral et al., 30 Dec 2025).

Taken together, these developments show that orbital altermagnetic phases are not a single mechanism but a symmetry-governed family of states. They range from orbital-order-induced spin altermagnets and anti-altermagnets, through pure orbital loop-current and sublattice-orbital orders, to topological, superconducting, and fractionalized descendants. What unifies them is the replacement of simple translation-based compensation by a point-group or layer-pattern compensation in which orbital structure is essential rather than ancillary.