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Stripe-Order Altermagnetism: Nematic Spin Splitting beyond the l-Wave Classification Zheng-Yang Zhuang 1 and Zhongbo Yan 1, ∗ 1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,

State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics, Sun Yat-sen University, Guangzhou 510275, China

(Dated: July 31, 2026)

arXiv:2607.17997v2 [cond-mat.mes-hall] 30 Jul 2026

Altermagnetism combines compensated magnetic order with nonrelativistic spin splitting, yet established mechanisms predominantly rely on spin-reversing rotations in Néel-order antiferromagnets. Here we establish stripe-order altermagnetism governed instead by a spin-reversing mirror, placing it outside the usual rotationbased l-wave classification. Using two-orbital models, we show that the interplay between stripe spin and orbital orders can yield a stripe altermagnet with mirror-constrained nematic spin splitting or a stripe anti-altermagnet with spin-degenerate bands. The latter can support ferroelectric-like electrical control of spin splitting in suitable buckled structures. Random-phase-approximation (RPA) calculations show that stripe-order altermagnetism is favored near half filling under strong hopping anisotropy. The spin-reversing mirror further enforces a purely transverse Drude spin current for an electric field parallel or normal to the mirror plane, while spin-resolved mirror and rotation probes distinguish this response from that of rotation-governed l-wave altermagnets.

Unconventional magnets can combine antiferromagnetic spin order with responses conventionally associated with ferromagnets [1–3]. Altermagnetism (AM) is a prominent example [4, 5]: it supports spin-split bands in a collinear magnet with vanishing net spin magnetization [5–7], thereby reconciling antiferromagnetic compensation with momentumdependent spin splitting. Such nonrelativistic spin splitting was anticipated in d-wave spin-density-wave states [8, 9] and spin-channel Pomeranchuk instabilities [10, 11], and was later recognized as a generic feature of a broad class of antiferromagnets in which a spin-reversing rotation or mirror, rather than inversion or translation, relates the opposite-moment sublattices [4, 5]. It has since been observed in CrSb [12–17], MnTe [18–21], and KV 2 Se 2 O [22–25]. Altermagnetism has also been linked to diverse phenomena across condensed matter physics [26, 27], most notably spintronics [28–33], topological phases [34–39], and unconventional superconductivity [40–44]. Altermagnetic spin splittings are commonly classified by the angular-harmonic degree l as p-wave (l = 1) [45–

l forms. These labels are assigned according to how the spin splitting transforms under spin-reversing rotations, which combine a real-space rotation with spin reversal and thereby constrain its angular dependence. This framework is prevalent in part because many widely studied altermagnets are Néel-order antiferromagnets (Néel AFMs), whose spin structures naturally accommodate such rotations [Fig. 1(b)]. In spin-space-group notation [5, 56–60], these symmetries are

AFMs and establish stripe-order altermagnetism as a realization of mirror-governed nematic altermagnetism, a class beyond the rotation-based l-wave framework. We use this term for altermagnetic states in which a spin-reversing mirror, but no spin-reversing rotation, relates the opposite-spin sectors and constrains their momentum-space spin splitting. Here “nematic” labels the symmetry of the spin-splitting texture rather than an additional order parameter. Symmetry analysis shows that orbital order provides a route to stripe AMs and electrically controllable stripe anti-altermagnets (stripe anti-AMs). RPA calculations identify the conditions favorable to stripe AM, while spin-current responses provide transport probes that distinguish mirror-governed nematic altermagnets

Symmetry and models for stripe AMs.– An altermagnet features spin-split bands yet zero net spin magnetization [4, 5]. This requires two conditions. First, the two magnetic sublattices with opposite moments must experience inequivalent local environments; hence spin-reversing inversion [C 2 ∥P] and translation [C 2 ∥t] symmetries must be absent. Second, the opposite-moment sublattices must still be related by a spinreversing symmetry. In a stripe AFM, this relation can be provided by a vertical mirror such as M x or M y . Within the two-dimensional stripe geometry considered here, a minimal criterion for realizing a stripe AM is the absence of [C 2 ∥P] (equivalently [C 2 ∥C 2z ]) and [C 2 ∥t], together with a preserved spin-reversing mirror.

Because stripe order often coexists with orbital order [70], we use the latter to break spin-reversing inversion [C 2 ∥P] and translation [C 2 ∥t]. Orbital-order-induced altermagnetism has also been explored in Néel AFMs [54, 72–74]. In momentum space, the Hamiltonian takes the general form

Here, s 0 is the identity matrix and s = (s x , s y , s z ) denotes the Pauli matrices in spin space. The terms H ele (k), H J , and H orb describe spin-independent hopping, stripe spin order, and orbital order, respectively. For the orbital-order mechanism considered here, a minimal model contains two sublattices, two orbitals, and two spin sectors, giving an eightband Hamiltonian. For concreteness, we consider a twodimensional square-lattice model with d xz and d yz orbitals [Fig. 2(a)],

The Pauli matrices τ and σ act in orbital and sublattice

stants a x and a y are set to unity throughout). These three terms describe, respectively, nearest-neighbor hopping along x, nearest-neighbor hopping along y, and next-nearestneighbor hopping, with corresponding amplitudes {t x0 , t x1 },

FIG. 2. Stripe-altermagnetic models and spin-resolved band structures. (a) Minimal orbital-order stripe-AM model and its reduced Brillouin zone. (b) Buckled stripe-spin/Néel-τ x -orbital model and its reduced Brillouin zone. The labels M x M and C 4z M denote the images of M under the corresponding operations, and G x = π is the reciprocal-lattice period along x. Panels (c) and (d) show spinresolved bands along the paths in (a) and (b), respectively. In (d), the spectra from left to right correspond to λ = 0, −0.8, +0.8. Finite λ activates the spin splitting, and λ → −λ reverses its pattern. Red and blue denote opposite spins, whereas black denotes spin-degenerate bands. The opposite-spin bands are related by the spin-reversing mirror [C 2 ∥M x ], rather than by the spin-reversing rotation [C 2 ∥C 4z ]. We set the orbital exchange fields δ x and δ g to 0.6 in the stripe-AM and stripe-spin/Néel-orbital models, respectively.

Jτ 0 σ z . Stripe orbital order with the same ordering vector is

and δ z τ z describe d xz ± d yz orbital antibonding/bonding and

Before orbital order develops (δ x = δ z = 0), the system preserves both [C 2 ∥P] and [C 2 ∥t]. Together with spinless time reversal, the former enforces spin degeneracy, while the latter does so directly. Orbital order removes these constraints, with the resulting magnetic state determined by the orbital channel. At finite δ x and δ z = 0, the d xz ± d yz order preserves [C 2 ∥M x ], which relates the two spin sectors and enforces vanishing net spin magnetization, while no remaining symmetry forces spin degeneracy. The system therefore realizes a stripe AM with mirror-constrained nematic spin splitting [75]. By contrast, finite δ z at δ x = 0 breaks [C 2 ∥M x ]. The spin sectors are then no longer related by symmetry, and a metallic system generally develops finite spin magnetization, corresponding to a weak ferrimagnetic state.

Figure 2(c) illustrates how mirror-constrained spin splitting in stripe AMs differs from its rotationally protected counterpart: opposite-spin bands are related by a mirror rather than a spin-reversing rotation. The resulting nematic spin splitting lies outside the rotation-based l-wave classification because no spin-reversing rotation assigns it a global l-wave label. This orbital-order realization also motivates a minimal fourband model, presented in End Matter.

3

Stripe-order anti-altermagnets.– The construction above corresponds to the first of three orbital configurations relative to stripe spin order: (i) stripe orbital order with the same ordering vector as the spin order, (ii) stripe orbital order with an ordering vector orthogonal to that of the spin order, and (iii) Néel orbital order. For τ x orbital-bonding order, cases (ii) and (iii) instead realize stripe-order anti-altermagnetic states [74] (also referred to as hidden-altermagnetic states [76]), whose hidden altermagnetic sectors can be revealed by an electric field. We focus below on case (iii); case (ii) is discussed in the Supplemental Material [77].

Combining Néel orbital order with stripe spin order enlarges the unit cell from two to four sublattices [Fig. 2(b)], giving a sixteen-band model after including spin. We denote this stripe (C-type)-spin/Néel (G-type)-orbital configuration by the superscript CG. The spin-independent hop-

whereas the stripe spin and Néel orbital orders are H J = Jσ z and H orb = δ g ρ z σ z τ x , respectively. The Pauli matrices σ and ρ act on the two sublattice degrees of freedom along x and y, respectively. As illustrated in Fig. 2(b), this model can be viewed in real space as two copies of the minimal stripeAM model in Fig. 2(a), with opposite orbital-order patterns, d xz ± d yz and d xz ∓ d yz , connected by a half-unit-cell translation along y. An electrostatic potential imbalance between the two copies is described by λρ z . Such an imbalance can arise from stripe charge order with an ordering vector orthogonal to the spin order. Alternatively, in the buckled geometry of Fig. 2(b), ferroelectric polarization can induce this imbalance, which can then be tuned by a vertical field with λ = eE z d/2 without breaking [C 2 ∥M x ]. Here e is the elementary charge and d is the vertical separation between the two copies.

At λ = 0, spin-reversing inversion [C 2 ∥P] relates the two copies and, together with spinless time reversal, enforces same-momentum spin degeneracy. The full band structure is therefore spin degenerate, even though each constituent copy is stripe altermagnetic. We refer to this state as a stripe anti-AM. Finite λ breaks [C 2 ∥P] and shifts the energies of the two copies in opposite directions, producing momentum-dependent spin splitting. The state thereby becomes altermagnetic because [C 2 ∥M x ] continues to enforce vanishing net spin magnetization. In the buckled geometry of Fig. 2(b), a vertical electric field controls λ. Reversing the field sends λ → −λ and reverses the spin-splitting pattern [Fig. 2(d)] without reversing the underlying spin or orbital order. The stripe-spin/Néel-orbital system thus realizes electrically switchable altermagnetism: the spin splitting vanishes at E z = 0, emerges at finite E z , and reverses with the field direction.

Having established these mechanisms, we next use RPA to identify the conditions favorable to stripe AM.

Spontaneous stripe AM.– Motivated by the occurrence of stripe order in correlated materials, we use RPA to determine the conditions under which interactions favor the spin and orbital orders required for stripe AM.

FIG. 3. RPA leading-instability diagram as a function of the chemical

The effective interaction strengths in the spin and orbital channels

k B T /t x0 = 0.04. The label “stripe AM” denotes points at which the leading spin and orbital channels match the stripe-AM configuration, whereas “stripe AFM” denotes a stripe AFM with spin-degenerate bands that does not belong to the anti-AM class. The label “others” collects parameter points for which at least one active channel has its global maximum outside the set {Γ q , X q , Y q , M q }. Parameters:

Before either order develops, the parent model has no enlarged unit cell and can be written as a single-sublattice twoorbital model (see Supplemental Material [77]). The ordering vector q s/o of the incipient spin or orbital order is identified by the momentum at which the corresponding suscep-

netic, stripe, stripe, and Néel order, respectively. Stripe spin order requires q s ∈ {X q , Y q }. Within the present setting, stripe AM requires q o = q s [case (i)], whereas stripe anti-

q o = M q [case (iii)]. Both phases considered here require τ x orbital-bonding order.

We introduce effective onsite interactions in the spin and orbital √ channels and define normalized orbital matrices τ̄ a = τ a / 2. The projected susceptibilities are evaluated on a discrete mesh D q covering 0 ≤ q x , q y ≤ π. After tracing over spin, the static bare susceptibility is

(τ x ), and orbital-polarization (τ z ) channels, with O ν =

tor of two accounts for spin degeneracy. For each channel, the

4

while U c,ν (q) = χ 0 1 (q) is the critical interaction strength.

ν Here U ν = U s for spin and U ν = U o for the two orbital channels. We identify the leading instability from the global

As shown in Fig. 3, the leading spin and orbital Stoner instabilities select the ingredients of stripe AM near half filling in the strongly anisotropic regime. Heuristically, the dominant x-direction hopping produces a quasi-one-dimensional dispersion and enhances nesting near q x = π, favoring antiferromagnetic modulation along x, consistent with the X q = (π, 0) stripe instability found numerically. As the hopping anisotropy is reduced, the stripe-AM region extends into the electron-doped regime. When only the effective onsite interactions are included, RPA preserves the locations of the baresusceptibility maxima, which generally coincide in the spin and orbital channels for the present bands. Neither stripe antiAM configuration is therefore selected. Nonlocal, orbitaldependent, or lattice-mediated interactions can modify this selection and may favor stripe anti-AM configurations in other regimes. The broad region labeled others reflects cases in which at least one leading instability occurs at a generally incommensurate wave vector outside {Γ q , X q , Y q , M q }, as is common in quasi-one-dimensional bands.

Spin-current response.– Nematicity controls not only the spin-splitting pattern of stripe AMs but also their transport response. In the absence of spin-orbit coupling, the preserved spinless time-reversal symmetry [ C̄ 2 ∥T ] forbids the intrinsic linear charge Hall response, leaving the dissipative Drude term as the leading dc contribution. For a fixed spin sector s, the Drude charge-conductivity tensor is [78, 79]

is the group velocity along the ith direction for the nth band in spin sector s.

The spin-reversing mirror constrains the two spin-resolved charge tensors according to

Accordingly, the two tensors take the form

(5)

(6)

spin conductivity and has the same units as the charge conductivity. An electric field parallel or normal to the mirror plane thus produces a purely longitudinal charge current together with a purely transverse Drude spin current.

FIG. 4. Drude spin-current response and symmetry probe for the mirror-governed stripe altermagnet. (a) Normalized longitudinal (L) and transverse (T) charge-equivalent spin conductances for E = E(cos θ, sin θ). Solid and dashed curves denote opposite signs. (b) Spin-resolved diagnostics: the C 4z -rotated difference is finite, while

gitudinal and transverse charge-equivalent spin responses per unit field are

lobed polar patterns [Fig. 4(a)]. This apparent fourfold structure was first predicted for Néel AM because of its inherent spin-reversing mirror symmetry [4]. However, spin-reversing mirror and rotation symmetries typically coexist in Néel AM, and the four-lobed polar pattern does not reveal whether the spin-reversing rotation is present. It therefore does not by itself distinguish mirror-governed nematic altermagnetism

The mirror- and rotation-based scenarios can instead be distinguished directly through the spin-resolved longitudinal

tation and mirror probes are

(8) (9)

The probes vanish when the spin sectors are related by C 4

tional probe extends directly to any candidate [C 2 ∥C nz ] symmetry by replacing the angular shift π/2 in Eq. (8) with 2π/n.

For the two-dimensional minimal orbital-order model, we use the representative parameters in Fig. 4, together with

at µ = 0.196 eV [77], while the C 4z probe in Fig. 4(b) reaches 4.843 × 10 −4 S and the mirror probe vanishes by symmetry. These 10 −4 -S signals should be accessible to state-of-the-art spin-sensitive transport measurements [28].

5 Discussion and conclusions.– We have introduced altermagnetism into stripe-order AFMs and established the resulting stripe-order altermagnetism as a realization of mirrorgoverned nematic altermagnetism beyond the rotation-based l-wave classification. A spin-reversing mirror relates the opposite-spin sectors and constrains their anisotropic spinsplitting pattern, while every crystallographic spin-reversing rotation is absent. For the two-orbital parent model, RPA identifies the stripe-AM ordering channels near half filling under strong hopping anisotropy. For electric fields parallel or normal to the spin-reversing mirror plane, symmetry enforces a purely transverse Drude spin current together with a purely longitudinal charge current. Combined with the spin-resolved mirror and rotation probes, this response distinguishes mirror-governed nematic altermagnets from rotation-

Stripe magnetism has been predicted or observed in many correlated systems, providing potential platforms for stripe AM when the required orbital order and spin-reversing mirror symmetry are present. Examples include stripe-ordered parent compounds of iron-pnictide superconductors [70], the (Eu-doped) high-T c cuprate La 1.8−x Eu 0.2 Sr x CuO 4 [67], the cobalt chalcogenide BaCoS 2 [80], and the perovskite manganite Sr 1−x Sm x MnO 3 [81]. Of particular relevance, Sr 1−x Sm x MnO 3 may host simultaneous stripe spin and orbital orders for 0.33 ≤ x ≤ 0.38 [81], whereas BaCoS 2 is predicted to combine stripe spin order with Néel orbital order within each ab plane [80].

This characteristic spin-reversing mirror also makes stripe AM a natural parent state for the recently proposed mixedparity altermagnetism [82]. In a mixed-parity altermagnet, the spin-polarized bands satisfy neither the even-parity relation

nevertheless, a spin-reversing mirror can enforce zero net spin magnetization and nematic spin splitting. By contrast, stripe AM preserves the spinless time-reversal symmetry [ C̄ 2 ∥T ] and therefore exhibits even-parity spin splitting. In a noncentrosymmetric stripe AM, applying circularly polarized light can break [ C̄ 2 ∥T ] and provide a route from even- to mixedparity spin splitting.

In summary, stripe AMs are distinguished from Néel AMs not only by their real-space magnetic order but also by their mirror-constrained nematic spin splitting and symmetryconstrained spin-current responses. Together, these results establish stripe AMs as a natural form of nematic altermagnetism—a mirror-based class beyond the conven-

Acknowledgments.– Z.-Y. Zhuang thanks Lun-Hui Hu for fruitful discussions on a closely related project. Z.-Y. Zhuang and Z. Yan acknowledge support from the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (Grant No. JYB2025XDXM403) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023B1515040023).

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8

End Matter Spin degeneracy from [C 2 ∥C 2z ] and spinless time reversal.– In the two-dimensional stripe geometry, among the

cases n = 1, 3 are incompatible with a finite collinear moment because an odd power generates pure spin reversal [C 2 ∥E], where E is the real-space identity. The n = 4, 6 operations rotate the ferromagnetic chains away from the fixed stripe direction and therefore do not preserve a single ordering vector q for stripe order. The only crystallographic spin-reversing rotation that can remain is thus [C 2 ∥C 2z ].

In the nonrelativistic collinear limit considered here, and in the absence of orbital time-reversal breaking, the Hamiltonian also preserves the spinless time-reversal symmetry [ C̄ 2 ∥T ]. This operation combines a 180 ◦ spin rotation with time reversal, leaving the conserved spin label unchanged while sending k → −k.

For a two-dimensional band, C 2z sends k → −k. Spinless time reversal and [C 2 ∥C 2z ] therefore imply, respectively,

ery momentum. Thus every state has an opposite-spin partner at the same momentum, and [C 2 ∥C 2z ] is incompatible with a spin-split stripe AM whenever [ C̄ 2 ∥T ] is preserved.

Minimal model for stripe AM.– Motivated by the construction in Fig. 2(a), we formulate a minimal model for stripe AM. Because the orbital and spin orders have the same ordering vector, the orbital order does not further enlarge the magnetic unit cell. At a phenomenological level, its redistribution of local electron density can therefore be encoded in effective hoppings, allowing us to suppress the explicit orbital degree of freedom. Retaining only sublattice and spin yields the four-band Hamiltonian

(E2) where σ and s act in sublattice and spin space, respectively. The momentum-dependent coefficients are

The tildes distinguish these phenomenological hopping amplitudes from those of the model in Eq. 2. The terms ∆ ′ 0 , ∆ ′ x , and ∆ ′ z describe sublattice-symmetric same-sublattice hopping, intersublattice hopping, and the difference between the same-sublattice hopping amplitudes on the two sublattices, respectively.

For fixed spin s = ±1, the corresponding block is written as

σ x exchanges the two sublattices, while the spin part of the

The first relation explicitly establishes [C 2 ∥M x ]. By con-

[C 2 ∥C 2z ] symmetry through the second relation. The Hamiltonian likewise lacks higher-fold spin-reversing rotations. For

The cross term 2sJ∆ ′ z (k) produces spin splitting at generic

change and sublattice-asymmetric hopping are required. The

splitting is therefore even under k → −k but odd under either momentum reflection. Because

it vanishes on k x = 0, ±π/2 and on k y = 0 and the equivalent Brillouin-zone boundaries. The k x nodes lie on M x -invariant lines and follow directly from the spin-reversing mirror. The k y nodes are likewise symmetry enforced, now by the mirror combined with spinless time reversal. The retained sine harmonic specifies only the interpolation between these nodal lines.

Equation (E5) has the same algebraic form as minimal models of Néel AMs [83]. Near Γ, the spin-splitting

(E7)

is locally d xy -like. However, this resemblance does not furnish a global l-wave label, because the full stripe-anisotropic Hamiltonian lacks [C 2 ∥C 4z ] symmetry. A mirror, rather than a rotation, relates the opposite-spin bands and constrains the complete spin-splitting pattern.