arXiv:2602.10641v1 [cond-mat.mtrl-sci] 11 Feb 2026
1 Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),
School of Physics, Beijing Institute of Technology, Beijing 100081, China The antiferroaxial state is emerging as an important ferroic order in condensed matter systems. Here, we establish antiferroaxial altermagnetism as a broadly prevalent, generic, and microscopically grounded multiferroic mechanism, in which antiferroaxial counter-rotating distortions both induce altermagnetism and enable its deterministic and reversible switching. Within a unified Landautheory and symmetry framework, we identify a symmetry-allowed trilinear invariant coupling the antiferroaxial order, the Néel vector, and the altermagnetic order, and derive general symmetry criteria for its occurrence. This coupling locks the induced altermagnetism to the antiferroaxial order, so that reversing the latter reverses the spin splitting and associated time-reversal-odd responses, such as anomalous Hall conductivity. We provide a practical spin group dictionary mapping Néelvector representations to the resulting d-, g-, and i-wave antiferroaxial altermagnetism, validate the mechanism with ligand-rotation tight-binding models and first-principles calculations, and identify many candidate materials by screening the MAGNDATA and C2DB databases. Our results elevate antiferroaxiality to a universal ferroic control knob for structurally programmable altermagnetic spintronics.
Introduction.— Altermagnetism has attracted widespread attention due to its coexistence of nonrelativistic spin-splitting bands and zero net magnetization, thereby combining key features of ferromagnets and antiferromagnets [1–40]. This unconventional property stems from rotational or mirror symmetries that relate the magnetic sublattices, rather than from translation or inversion symmetries, as illustrated in Fig. 1(a). In the limit of vanishing spin-orbit coupling (SOC), the Néel vector N of altermagnets transforms as Γ AM ⊗ Γ sA , where
(1D) irreducible representation (IR) of the little group at the Γ point, and Γ sA denotes the axial-vector IR of the spin-rotation group. The hallmark altermagnetic nonrelativistic spin splitting
is characterized by a mag-
components [41, 42]. The brackets [· · · ] indicate the symmetric-traceless projection over µ 1 · · · µ ℓ , so that O (ℓ) transforms as Γ ℓ ⊗ Γ sA , with Γ ℓ the rank-ℓ spatial IR. Recent studies have reported that these altermagnetic multipoles can couple to classical multiferroic orders [43–50] , such as the ferroelectric and ferroelastic ones, suggesting a path toward altermagnetic multiferroicity.
In addition to conventional ferromagnetic, ferroelectric, and ferroelastic orders, there is a new type of significant ferroic order, the ferroaxial order [51–59]. Ferroaxiality is characterized by the rotational distortion of structural units, represented by the axial vector A [60], whose magnitude is parameterized by the rotation angle θ. The ferroaxial order transforms according to the axial-vector IR Γ A and is invariant under inversion and time reversal symmetry. Analogously, the antiferroaxial order is characterized by structural units exhibiting counter-rotating distortions of equal magnitude but op-
posite directions along a shared axis [51], as illustrated in Fig. 1(b). This state is characterized by a staggered order parameter G = A a − A b , where A a and A b denote the local ferroaxial vectors of sublattices a and b, respectively. Antiferroaxial order commonly occurs in many 2 systems, such as perovskites and transition metal trifluorides [61–65]. Unlike the ferroelectric order, the ferroaxial and antiferroaxial orders do not require inversion symmetry breaking and can couple to physical quantities such as strain, optical fields, and piezoresistance [56–59].
In this Letter, we propose antiferroaxial altermagnetism, a multiferroic state in which altermagnetism is generated and reversibly switched by the antiferroaxial order. Using symmetry analysis and Landau theory, we demonstrate a symmetry-allowed trilinear coupling among the antiferroaxial order G, the Néel order N, and magnetic multipoles O (ℓ) , and establish the general symmetry criteria governing this interaction. This coupling dictates that reversing the antiferroaxial order inverts the spin splitting, enabling multiferroic control. Furthermore, Table II classifies the altermagnetic spin point groups and their associated Γ AM , serving as a practical guide for determining the spin splitting type of antiferroaxial altermagnetism. We illustrate antiferroaxial altermagnetism via tight-binding models incorporating ligand rotations, confirming its applicability to generate and control d-, g-, and i-wave altermagnetism. Using first-principles calculations combined with screening the MAGNDATA [66] and C2DB [67, 68] databases, we identify many candidate antiferroaxial altermagnets (Table I). Taking FeF 3 as an example, we demonstrate that reversing the antiferroaxial order not only switches the spin splitting but also inverts the time-reversal-odd responses, such as the anomalous Hall conductivity (AHC), establishing a novel mechanism for structural control of spintronics.
Landau theory.— In Landau theory, an antiferroaxial order G transforms as Γ G (q) ⊗ Γ A in the parent space group G 0 , where Γ G (q) is the IR of the little group at the propagation vector q that encodes the staggered sublattice pattern and Γ A denotes the axial-vector representation. Similarly, in the zero SOC limit, the Néel vector
pling among G, N, and O (ℓ) requires that the product of their representations contains the identity representation. Since the spin-rotation sector satisfies Γ s 1 ⊂ Γ sA ⊗ Γ sA , where Γ s 1 is the identity representation in spin-rotation space, the invariance condition reduces to the spatial sector only,
This implies that the magnetic multipole O (ℓ) can be generally induced by the coexistence of the antiferroaxial order and the Néel vector. Moreover, translational invariance of the coupling enforces momentum conservation, q ′ = −q. Equation (1) identifies all symmetry-allowed O (ℓ) ; specifically, the lowest rank ℓ dictates the type (d-, g-, or i-wave) of the induced altermagnetism. However, the spin-splitting type (planar or bulk) is established only upon the condensation of G. Once G condenses and reduces the symmetry to the isotropy subgroup G, the Néel vector’s representation in G is obtained by subducing the parent representation, i.e., Γ N (q ′ ) ↓ G (see Supplemental Material (SM) [69]). Equation (1) can be stated equivalently as follows: a system is an antiferroaxial altermagnet if and only if, after the antiferroaxial order G condenses and lowers the symmetry to the isotropy subgroup G, the subduced representation of the Néel vector, Γ N (q ′ ) ↓ G, contains a 1D IR that matches one of the Γ AM listed in Table II, which maps each Γ AM to its corresponding splitting type and collinear spin point group.
The coupling dictates that reversing G at fixed N necessarily enforces a sign reversal of the magnetic multipole O (ℓ) and the associated spin splitting, thereby establishing the multiferroic mechanism in which a nonmagnetic antiferroaxial order deterministically controls magnetic responses. Unlike ferroelectric orders, the antiferroaxial order does not necessarily break inversion symmetry, allowing this multiferroic control to operate even in nonpolar, centrosymmetric systems.
Model.— To clarify the microscopic mechanism, we perform the explicit symmetry analysis and construct tight-binding models that incorporate the ligand degrees of freedom. This allows us to directly map the antiferroaxial order onto the electronic structure. The model is defined on a lattice with magnetic atoms on sublattices a and b, bridged by nonmagnetic ligands whose rotational distortion encodes the antiferroaxial order [see Fig. 2]. The corresponding Hamiltonian reads (see SM [69])
(2)
Here, d † m (c † i ) denotes the creation operator for an electron on magnetic site m (ligand site i). The first term represents the hopping t M between nearest-neighbor magnetic sites. The second and third terms describe the magnetic-ligand hybridization V mi (θ) and the ligandligand hopping t ij (θ), respectively. Crucially, the hopping amplitudes such as t 1/2 and V 1/2 vary with the rotation angle θ due to changes in corresponding bond lengths [Figs. 2(a1)-(c1)]. This structural dependence is incorporated via Harrison’s scaling law t(r) ∝ r −2 [70], where the explicit forms of V mi (θ) and t ij (θ) are given in SM [69]. The sums over mi and ij run over the symmetry-inequivalent pathways (V 1/2 and t 1/2 ) illustrated in Fig. 2. The final term introduces the local antiferromagnetic exchange J, where σ z is the spin operator and η m = ±1 encodes the staggered Néel order.
Figures 2(a1)-(a3) show the geometry, band structure, and spin splitting of the d-wave antiferroaxial altermagnetic model. The antiferroaxial order is parameterized by the ligand rotation angle θ, as illustrated in Fig. 2(a1).
3 (a1) (b1)
y y x 1 1
vector Γ 2 (A ′ 2 ), corresponding to a planar i-wave spin splitting in 6̄m2 (Table II). The common model parameters are J = t
The parent phase belongs to space group P 4/mmm. As a representative scenario, we consider the transitions at
and Γ i follow the Bradley-Cracknell convention [71]. In the P 4/mmm group, the rank-2 multipole representa-
Table S1 in SM [69]), so the representation constraint in Eq. (1) is satisfied and a symmetry-allowed trilinear invariant exists. When G z condenses, the system symmetry is reduced to P 4/mbm [Fig. 2(a1)], which has crystallographic point group 4/mmm. The propagation vector q is folded to the Brillouin-zone center, and the Néel vector transforms as N ∼ B 2g (see details in SM [69]). Table II identifies this IR in point group 4/mmm corresponding to a d-wave spin splitting, with the associated altermagnetic spin point group 1̄ 4/m 1̄ mm. We use Litvin notation for spin point groups here [72]. The numerical results of our generic TB model with θ = 20 ◦ also show the same d-wave spin splitting. Reversing θ, i.e., reversing G, is symmetry-equivalent to the operation [E||C 4z ], thereby reversing the sign of the spin splitting (see details in SM [69]). Here [E||C 4z ] is a spin-group operation, where E denotes the identity operation in spin space and C 4z denotes a four-fold rotation in real space.
For the g-wave case, the parent group is also P 4/mmm. In contrast to Fig. 2(a), here the antiferroaxial order and the Néel order share the same spatial
The corresponding bilinear combination transforms as
Table S1 in SM [69]). After G z condenses, the symmetry is reduced to P 4/mbm [Fig. 2(b1)] and the Néel vector transforms as N ∼ A 2g , corresponding to planar g-wave altermagnetism (Table II). The geometry, band structure, and spin splitting are shown in Figs. 2(b1)-(b3), with parameters θ = 20 ◦ . Reversing G is symmetry-
splitting reversal (see details in SM [69]).
Figures 2(c1)-(c3) show the i-wave antiferroaxial altermagnetic model with parent group P 6/mmm. Both antiferromagnetic and antiferroaxial transitions occur at the Γ point, transforming as N ∼ Γ − 3 and G z ∼
in SM [69]). After G z condenses, the space group reduces to P 6̄m2, and the Néel vector transforms as A ′ 2 , which identifies the system as a planar i-wave altermagnet (Table II), as shown in Figs. 2 (c2) and (c3) with θ = 30 ◦ . Reversing the antiferroaxial order is symmetry-equivalent to [E||m 110 ], which also reverses the spin splitting (see details in SM [69]).
Antiferroaxial altermagnets.— To validate our theoretical framework and demonstrate the ubiquity of the proposed mechanism, we performed a comprehensive screening across the MAGNDATA [66, 73] and C2DB [67, 68] databases. Guided by the above theoretical analysis, we identified a series of candidate materials hosting antiferroaxial altermagnetism, as seen in Table I. In the following, we present two representative examples: the two-dimensional (2D) g-wave altermagnet MnS 2 [74] and the three-dimensional (3D) experimentally synthesized d-wave altermagnet La 2 NiO 4 [75], confirming their antiferroaxial altermagnetism via density functional theory (DFT) calculations.
Monolayer MnS 2 [74] is built from Mn-S 4 tetrahedra possessing S 4 symmetry [Fig. 3(a)]. Its parent phase 4
(c) (d)
Y M Γ X + 0 k x Spin splitting k y
antiferroaxial altermagnets
La 2 NiO 4 MnA 2 (A=S,C), KMnF 3 ,Ca 2 MnO 4
TABLE. I. The candidate antiferroaxial altermagnets and the corresponding altermagnetic types.
k z
k x
k y - FIG. 3. Antiferroaxial altermagnets. The blue/red spheres and green arrows denote opposite spins and antiferroaxial orders, respectively. (a, c) Monolayer MnS 2 (parent group P 4̄m2), where Néel (M 1 ) and antiferroaxial (M 2 ) orders couple with a O (4) magnetic multipole in the Γ 2 (A 2 ) channel (Table S1 in SM [69]). The symmetry lowers to space group
wave spin splitting (Table II). (b, d) Bulk La 2 NiO 4 (parent group I4/mmm), where Néel (X 1 + ) and antiferroaxial (X 3 + ) orders couple with the Γ + 5 (E g ) magnetic multipole channel. The symmetry lowers to space group P ccn with Néel vector Γ + 4 (B 3g ), corresponding to d yz -wave splitting (Table II). The insets in the band structures (c, d) illustrate the spin splitting within the Brillouin zone.
belongs to the space group P 4̄m2. The DFT calculations [74] reveal that both antiferroaxial and antiferromagnetic instabilities emerge at the M point (q = (π, π, 0)), leading to an ordered state where the MnS tetrahedra exhibit a substantial rotational distortion of θ ≈ 23.4 ◦ . Symmetry analysis indicates that the antiferroaxial and antiferromagnetic order parameters transform according to the irreducible representations
of the O (4) (see Table S1 in SM [69]). After G z condenses,
cating a g-wave spin splitting (Table II), which is corroborated by the DFT results shown in Fig. 3(c).
The 3D Ruddlesden-Popper compound La 2 NiO 4 [75] (parent space group I4/mmm) consists of corner-sharing Ni-O 6 octahedra [Fig. 3(b)]. Symmetry-mode analysis indicates concomitant antiferroaxial and antiferromagnetic instabilities at the X point [76, 77] (q = (π, π, 0)), yielding an antiferroaxial rotation pattern with θ ≈ 6.3 ◦ together with antiferromagnetic order, i.e., G 110 ∼ X 3 +
the E g channel of the magnetic multipole O (2) , implying an altermagnetic state. Upon condensation, G 110 lowers the crystallographic symmetry to I4 2 /ncm, and the N further reduces it to space group P ccn with point group mmm. In this state, the Néel order transforms as N ∼ Γ + 4 (B 3g ), corresponding to spin point group 1̄ mm 1̄ m with d-wave altermagnetism (Table II), confirmed by the band structure calculations [Fig. 3(d)].
Antiferroaxial Multiferroicity.— We formulate the Landau theory in terms of the Landau free energy to demonstrate how the antiferroaxial order switches the spin splitting and the associated physical properties, such as AHC, in altermagnets. As an example, we consider the antiferroaxial multiferroicity in the bulk g-wave altermagnet FeF 3 [63]. FeF 3 is composed of corner-sharing FeF 6 octahedra [Fig. 4(a)]. Starting from the parent space group P m 3̄m, it exhibits antiferromagnetic and antiferroaxial instabilities at the R point (q = (π, π, π)).
a T 1g spatial channel (see Table S1 in SM [69]), the representation constraint in Eq. (1) is satisfied, implying a symmetry-allowed trilinear coupling and a g-wave altermagnetism whose planar or bulk character is determined by the orientation of G.
Based on Landau theory, the free energy can be written as (see the SM [69]):
metries. Here N 2 ≡ N · N and G 2 ≡ G · G. The remaining terms F O , F ε , and F εG denote the magnetic multipole, elastic, and strain-antiferroaxial coupling contributions, respectively. Eliminating the secondary order parameters by their minimization conditions and substituting the equilibrium values back into F, we obtain an effective Landau free energy for G,
(4) where the specific expressions of the effective renormalized coefficients α eff , β 1,eff and β 2,eff are given in the 5
FIG. 4. Multiferroicity with antiferroaxial switchable spin splitting and anomalous Hall conductivity in altermagnets. (a) Top and side views of the crystal structure of FeF 3 . Red and blue spheres denote Fe atoms with opposite magnetic moments, while gray spheres represent F atoms. The green curved arrows indicate the antiferroaxial rotation of the FeF 6 octahedra, parameterized by the angle θ. (b) Band structure for the lowest-energy configuration at θ = 12 ◦ . The inset shows the 3D Brillouin zone with the bulk g-wave spin splitting. (c) Total energy as a function of the rotation angle θ, revealing a double-well potential with ground-state minima at θ ≈ ±12 ◦ . The insets illustrate that reversing the structural rotation direction (top) switches the sign of the spin splitting (bottom). The effective Landau free en-
structural distortion G (in degrees) and F eff is in meV per unit cell. (d) Anomalous Hall conductivity, calculated as a function of chemical potential at the equilibrium angles θ = ±12 ◦ , is switchable by antiferroaxial order.
SM [69]. The sign of the anisotropy parameter β 2,eff determines the ground state orientation of the antiferroaxial order. When β 2,eff < 0, the antiferroaxial vector aligns along the (111) direction (G x = G y = G z ), stabilizing the bulk g-wave spin splitting described by
observed in FeF 3 [Fig. 4(b)], NiF 3 , CoF 3 , and LaCrO 3 . Conversely, β 2,eff > 0 favors alignment along a principal axis (e.g., x, y, or z), leading to a planar g-wave splitting, as observed in KMnF 3 and Ca 2 MnO 4 [69].
The magnitude of the antiferroaxial order is quantified by the rotation angle θ of the Fe-F 6 octahedra. The DFT calculations reveal a double-well potential for the total energy as a function of θ [Fig. 4(c)], with minima located at θ = ±12 ◦ . The energy barrier is approximately ∆E ≈ 30 meV. Although the Néel vector remains fixed, the two antiferroaxial states (±θ) are related by the
Consequently, reversing the antiferroaxial order necessitates a reversal of the spin splitting, establishing FeF 3 as an antiferroaxial multiferroic altermagnet. Furthermore, in the consideration of the SOC, the two antiferroaxial states are related by the T τ symmetry. Since the AHC is odd under time-reversal, this symmetry relation dictates that reversing the antiferroaxial order can simultaneously switch the sign of the AHC, and more generally of any time-reversal-odd response, which is confirmed by the calculations shown in Fig. 4(d).
Discussion.— In this Letter, we demonstrate a novel class of multiferroic altermagnetism, termed antiferroaxial altermagnetism, in which altermagnetism is coupled with and tunable by antiferroaxiality. Based on symmetry analysis and Landau theory, we elucidate the general symmetry criteria governing the trilinear coupling between antiferroaxial, antiferromagnetic, and altermagnetic orders. We substantiate this theory by constructing general ligand-rotation tight-binding models for d-, g-, and i-wave antiferroaxial altermagnetism, confirming the coupling between antiferroaxiality and altermagnetism. By screening the MAGNDATA and C2DB databases combined with ab initio calculations, we identified a series of material candidates, exemplifying our theory with
termagnet La 2 NiO 4 . Furthermore, using FeF 3 as a prototype, we quantified a switchable energy barrier of 30 meV, demonstrating that reversing the antiferroaxial order deterministically switches the spin splitting and timereversal-odd responses such as AHC.
Our results establish antiferroaxiality as a key ferroic degree of freedom enabling the emergence and active control of altermagnetism. Recent progress in ferroaxial materials has demonstrated deterministic switching of rotational order using circularly polarized terahertz pulses via nonlinear phonon excitation [56, 57]. Motivated by this progress, and considering that antiferroaxial order arises from staggered rotational lattice modes, the analogous dynamical protocols could be extended to manipulate antiferroaxiality. Importantly, in the present framework, reversing the antiferroaxial order not only flips the altermagnetic spin splitting and associated transport responses, but also constitutes a deterministic reversal of the underlying structural order itself. Furthermore, we show that antiferroaxiality can be efficiently tuned and switched on or off by uniaxial strain, and identify KMnF 3 as a concrete material realization (see SM for details [69]). This work would establish a new paradigm for altermagnetic spintronics, enabling structural-magnetic co-engineering for the active control of spintronic devices even in centrosymmetric materials.
Acknowledgments.— The work is supported by the Science Fund for Creative Research Groups of NSFC (Grant No. 12321004), the NSF of China (Grant No. 12374055), and the National Key R&D Program of China (Grant No. 2020YFA0308800).
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END MATTER In the zero SOC limit, a collinear altermagnet is characterized by a Néel vector N that transforms as Γ AM ⊗Γ sA . Here Γ AM is a nontrivial 1D IR of the crystallographic point group and, when inversion is present, it must be inversion-even, while Γ sA denotes the axial vector IR in spin-rotation space. This transformation property implies that neither T τ nor IT enforces spin degeneracy in the zero SOC limit, allowing a momentum-dependent spin splitting. Given the altermagnetic spin point group, we determine whether the ordered phase is altermagnetic and identify whether its spin splitting belongs to the dwave, g-wave, or i-wave type. In the present work, once the antiferroaxial and Néel orders select a specific Γ AM through the symmetry-allowed coupling, Table II provides a direct lookup of the corresponding spin splitting type. All IRs with allowed nontrivial spin splitting are listed in Table II. A more detailed classification, including the associated lowest-order nodal functions, is provided in Table S3 in SM [69]
It is worth noting that, in the parent group G 0 , the bilinear product transforms as GN ∼ Γ G (q) ⊗ Γ A ⊗ Γ N (q ′ ) ⊗ Γ sA . After G condenses, the symmetry is lowered to the isotropy subgroup G, and the order parameter G transforms as the identity representation, that is
according to the altermagnetic IR Γ AM ⊗ Γ sA , namely
can be stated equivalently as follows: a system realizes antiferroaxial altermagnetism if and only if, upon condensation of the antiferroaxial order, the irreducible representation of the Néel vector in the resulting isotropy subgroup belongs to one of the Γ AM listed in Table II.
9 TABLE. II. Symmetry classification of antiferroaxial altermagnetism by collinear spin point groups (SPG). The table lists crystallographic point groups (PG) and their symmetryallowed collinear spin point groups. Γ AM denotes the irreducible representation of the Néel vector. The spin splitting
and β ∈ {1, 2} denote the distinct irreducible representations corresponding to the same spin point group. Mulliken symbols are used for the irreducible representations of point groups, while spin point groups follow Litvin notation [72]. Groups with no allowed non-trivial spin splitting are omitted.
PG SPG (Γ AM , Type) PG SPG (Γ AM , Type)
Supplementary material for “Antiferroaxial altermagnetism”
School of Physics, Beijing Institute of Technology, Beijing 100081, China
I. POSSIBLE MAGNETIC MULTIPOLES IN ANTIFERROAXIAL ALTERMAGNETISM
II. CLASSIFICATION OF ANTIFERROAXIAL ALTERMAGNETISM IN TERMS OF SPIN POINT
GROUP
ISOTROPY SUBGROUP AND IRREP SUBDUCTION
IV. LIGAND-ROTATION TIGHT-BINDING MODELS
5 (a1) (a2) (a3) (a4)
A. d-wave antiferroaxial altermagnetic tight-binding model
B. g-wave antiferroaxial altermagnetic tight-binding model
7 C. i-wave antiferroaxial altermagnetic tight-binding model
