D-Wave Altermagnets

• D-Wave altermagnets are collinear magnets with zero net magnetization, exhibiting a d-wave spin splitting that reverses sign under a 90° rotation.
• Minimal two-band Hamiltonians incorporating d-wave spin-splitting terms capture experimental observations such as anisotropic Fermi surfaces and nodal line behavior.
• Experimental studies reveal unique optical, transport, and topological signatures, indicating strong potential for spintronics and novel quantum device applications.

D-wave altermagnets are collinear magnetic systems with zero net magnetization and momentum-dependent spin splitting, in which the spin splitting changes sign under a $90^\circ$ rotation and commonly takes the form $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$ or $\lambda (k_x^2-k_y^2)$ (Weber et al., 2024, Yang et al., 10 Sep 2025). They are distinct from ferromagnets, which exhibit a uniform spin splitting, and from conventional antiferromagnets, in which each $\mathbf k$ state is doubly Kramers-degenerate once spin-orbit coupling is neglected (Weber et al., 2024). Across current theory and experiment, d-wave altermagnets are treated as a class of materials in which spin-crystal symmetry coupling produces alternating spin polarization in reciprocal space while the total moment remains compensated, enabling transport, optical, multipolar, and collective phenomena that have no direct counterpart in conventional collinear antiferromagnetism (Fu et al., 30 Dec 2025, Zarzuela et al., 2024).

1. Symmetry, order parameter, and defining distinction

A recurring defining statement is that altermagnets are collinear magnets that, like antiferromagnets, have zero net moment, yet, like ferromagnets, break time-reversal symmetry (Fu et al., 30 Dec 2025). In d-wave altermagnets, the two spin sublattices are not related by a pure translation or inversion; instead, they are related by a combined nonrelativistic spin-group operation such as $[C_2\|C_{4z}]$, or, in magnetic-space-group language, by a combined rotation and time reversal such as $C_4\Theta$ (Fu et al., 30 Dec 2025, Wei et al., 2024). This symmetry enforces a momentum-dependent spin splitting that reverses sign under $90^\circ$ rotation.

The standard d-wave form factor is written either as

$f_d(\mathbf k)=\cos k_x-\cos k_y$

or, near $\Gamma$, as

$\psi_d(\mathbf k)\propto k_x^2-k_y^2.$

Under $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$0, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$1 changes sign, while under $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$2 it is invariant; this sign reversal encodes the d-wave character of the altermagnetic order (Fu et al., 30 Dec 2025). In La$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$3O$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$4Mn$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$5Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$6, symmetry analysis in $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$7 assigns the altermagnetic order parameter to the $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$8 irreducible representation with simplest basis function $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$9 (Wei et al., 2024). In CsV$\lambda (k_x^2-k_y^2)$0Se$\lambda (k_x^2-k_y^2)$1O, the same sign-changing structure is described as a $\lambda (k_x^2-k_y^2)$2 form factor $\lambda (k_x^2-k_y^2)$3 (Fu et al., 30 Dec 2025). This suggests that the essential invariant across notational conventions is the alternating sign structure under $\lambda (k_x^2-k_y^2)$4 rather than a single universal label.

Several works formulate the d-wave order through a spin-resolved expectation value. One compact expression is

$\lambda (k_x^2-k_y^2)$5

with $\lambda (k_x^2-k_y^2)$6 (Fu et al., 30 Dec 2025). Another is the momentum-space spin density constraint

$\lambda (k_x^2-k_y^2)$7

which makes explicit that the net magnetization still vanishes after Brillouin-zone integration (Wei et al., 2024). In this sense, d-wave altermagnetism is not defined by a uniform spin moment, but by a symmetry-protected sign-alternating spin texture in momentum space.

2. Minimal Hamiltonians and reciprocal-space structure

The minimal single-particle description is a two-band Hamiltonian with a spin-independent dispersion and a d-wave spin-splitting term,

$\lambda (k_x^2-k_y^2)$8

with $\lambda (k_x^2-k_y^2)$9 or $\mathbf k$0 (Wei et al., 2024, Jiang et al., 2024). For La$\mathbf k$1O$\mathbf k$2Mn$\mathbf k$3Se$\mathbf k$4, one $\mathbf k$5 form is

$\mathbf k$6

giving

$\mathbf k$7

The spin splitting vanishes along the nodal planes $\mathbf k$8 and reaches extrema along the principal axes (Wei et al., 2024).

A continuum model used for nonlinear opto-magnetic response is

$\mathbf k$9

whose two bands $[C_2\|C_{4z}]$0 are fully spin-split, yet the net magnetization

$[C_2\|C_{4z}]$1

remains zero (Yang et al., 10 Sep 2025). A lattice realization for KRu$[C_2\|C_{4z}]$2O$[C_2\|C_{4z}]$3 is

$[C_2\|C_{4z}]$4

with

$[C_2\|C_{4z}]$5

and parameters $[C_2\|C_{4z}]$6, $[C_2\|C_{4z}]$7, $[C_2\|C_{4z}]$8 (Yang et al., 10 Sep 2025).

For metallic KV$[C_2\|C_{4z}]$9Se$C_4\Theta$0O, the band structure is modeled as

$C_4\Theta$1

with

$C_4\Theta$2

$C_4\Theta$3

The eigenvalues

$C_4\Theta$4

yield spin-split bands along $C_4\Theta$5–X and degeneracy along the nodal lines $C_4\Theta$6 (Jiang et al., 2024).

A separate tight-binding formulation emphasizes topological structure in two dimensions:

$C_4\Theta$7

with

$C_4\Theta$8

In that model, the direct gap closes at the critical point

$C_4\Theta$9

and for $90^\circ$0 the system enters a topological nodal semimetal with Dirac points, Berry-curvature singularities, and pseudospin-texture winding (Calixto, 4 Feb 2026).

3. Material platforms and experimental characterization

Current literature spans correlated insulators, metallic layered vanadium oxychalcogenides, rutile oxides, and atomically probed candidate systems.

Material	Reported status	Key reported signatures
La$90^\circ$1O$90^\circ$2Mn$90^\circ$3Se$90^\circ$4	correlated insulating layered d-wave altermagnet	$90^\circ$5; full insulating gap $90^\circ$6; spin splitting up to $90^\circ$7; robust 2D AFM fluctuations above $90^\circ$8 (Wei et al., 2024)
KV$90^\circ$9Se$f_d(\mathbf k)=\cos k_x-\cos k_y$0O	metallic room-temperature d-wave altermagnet	highly anisotropic spin-polarized Fermi surfaces; SDW below $f_d(\mathbf k)=\cos k_x-\cos k_y$1 (Jiang et al., 2024, yan et al., 30 Apr 2025)
CsV$f_d(\mathbf k)=\cos k_x-\cos k_y$2Se$f_d(\mathbf k)=\cos k_x-\cos k_y$3O	candidate material with atomic-scale visualization	unidirectional electronic patterns tied to magnetic domain walls and spin defects; elliptical charging rings; SDW gap $f_d(\mathbf k)=\cos k_x-\cos k_y$4 (Fu et al., 30 Dec 2025)
RbV$f_d(\mathbf k)=\cos k_x-\cos k_y$5Se$f_d(\mathbf k)=\cos k_x-\cos k_y$6O	2D d-wave altermagnet with spin-texture locking	spin-lattice, spin-scattering, spin-momentum, and spin-stripe locking visualized by SP-STM and QPI (Mu et al., 20 Apr 2026)
RuO$f_d(\mathbf k)=\cos k_x-\cos k_y$7	prototypical d-wave altermagnet in optical studies	optical analogue of a spin splitter effect; pump-polarization-dependent persistent optically excited electronic spin polarization (Weber et al., 2024)

In La$f_d(\mathbf k)=\cos k_x-\cos k_y$8O$f_d(\mathbf k)=\cos k_x-\cos k_y$9Mn$\Gamma$0Se$\Gamma$1, the parent crystal is body-centered tetragonal $\Gamma$2, neutron diffraction finds G-type antiferromagnetic order at $\Gamma$3, and the resulting magnetic space group is $\Gamma$4 (Wei et al., 2024). DFT+$\Gamma$5 with $\Gamma$6 on Mn-$\Gamma$7 predicts a full insulating gap larger than $\Gamma$8 and momentum-dependent spin splitting up to $\Gamma$9 along $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$0–X, reversing sign along $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$1–Y, with nodal planes at $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$2 (Wei et al., 2024). Magnetometry shows a sharp kink at $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$3, while neutron pair distribution function analysis shows a 2D short-range magnetic component that persists above the Néel temperature (Wei et al., 2024).

KV$\psi_d(\mathbf k)\propto k_x^2-k_y^2.$4Se$\psi_d(\mathbf k)\propto k_x^2-k_y^2.$5O is reported as a metallic room-temperature d-wave altermagnet with crystal space group $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$6 and a zero-temperature collinear order above $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$7 described by the spin-space symmetry operation $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$8 (Jiang et al., 2024). Below $\psi_d(\mathbf k)\propto k_x^2-k_y^2.$9, a secondary spin-density wave develops, a small spin canting reduces the magnetic symmetry, and DFT shows band-degeneracy lifting, Fermi-surface reconstruction, and a magnetic-breakdown mechanism that accounts for contrasting Hall resistivity relative to the C-type AFM state (yan et al., 30 Apr 2025).

Direct real-space evidence is reported for CsV$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$00Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$01O and RbV$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$02Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$03O. In CsV$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$04Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$05O, STM at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$06 resolves strip-like domain walls along $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$07, defect-bound in-gap resonances at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$08 and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$09, quasi-1D charge modulation of wavelength $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$10, and elliptical charging rings elongated along either $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$11 or $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$12 (Fu et al., 30 Dec 2025). In RbV$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$13Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$14O, SP-STM with a field-switchable Cr tip at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$15 reveals sublattice-resolved out-of-plane spin contrast, spin-dependent QPI anisotropy, and a long-period stripe modulation with $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$16, interpreted as quadruple spin-texture locking (Mu et al., 20 Apr 2026).

4. Optical, nonlinear, and opto-magnetic responses

A central nonlinear response is the inverse Cotton–Mouton effect (ICME), in which monochromatic linearly polarized light induces a static magnetization,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$17

For a d-wave altermagnet with combined spin-inversion $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$18 and fourfold rotation $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$19 symmetry,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$20

and for in-plane polarization angle $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$21,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$22

When a mirror plane makes $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$23, the result reduces to

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$24

so the induced magnetization is locked to the Néel vector and has $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$25 periodicity in the pump polarization angle (Yang et al., 10 Sep 2025).

For the KRu$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$26O$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$27 parameter set, numerical evaluation gives a dip in $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$28 at terahertz frequencies and a sign change at higher $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$29, with $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$30 (Yang et al., 10 Sep 2025). Choosing $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$31 gives $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$32, and for $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$33, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$34, and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$35 one finds $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$36, implying

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$37

For $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$38 order, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$39 with maxima at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$40 and zero crossings at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$41; for $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$42 order, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$43 (Yang et al., 10 Sep 2025).

An experimentally realized optical analogue of the spin-splitter effect was reported for ultrathin RuO$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$44 films. In the ab-initio description, the optically excited spin polarization obeys

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$45

with zero crossings at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$46 (Weber et al., 2024). Pump–probe measurements on a $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$47 epitaxial RuO$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$48(001) film used a $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$49, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$50 pump and a $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$51 polar-MOKE probe. At $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$52, the Kerr signal showed a clear $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$53 periodicity with a maximum at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$54 and opposite sign at $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$55; by $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$56, the $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$57 modulation had vanished (Weber et al., 2024).

Third-order photoconductivity has also been derived from a microscopic multi-orbital tight-binding model. In that framework, the third-order injection and shift currents are determined solely by the quantum metric and quantum connection and are free from Berry-curvature contamination (Zhang, 14 Apr 2026). In the ideal limit $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$58, closed-form analytical expressions are obtained; for finite $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$59, perturbative analytical results agree well with numerical calculations up to $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$60 (Zhang, 14 Apr 2026). In the clean limit, the third-order injection current exceeds the shift current by $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$61, and in the ideal limit the sublattice–orbital–spin locking yields $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$62 spin-polarized third-order currents (Zhang, 14 Apr 2026).

5. Transport, multipoles, strain, and collective modes

Mesoscopic transport theory predicts a spin-polarized diffusive contribution to the effective Hamiltonian that has no counterpart in conventional antiferromagnetism and is responsible for the spin-splitter effect (Zarzuela et al., 2024). In the hydrodynamic limit, the spin current contains a term proportional to the orthogonal charge current and polarized along the Néel vector, producing a transverse spin current in a fully nonrelativistic setting (Zarzuela et al., 2024). The same framework yields a distinctive spin-transfer torque,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$63

and predicts domain-wall motion driven by transverse charge currents (Zarzuela et al., 2024).

Several transport responses are formulated in terms of higher multipoles. One line of work treats d-wave altermagnets as systems whose order parameters are magnetic octupoles, and shows that octupoles injected from a heavy metal generate torque on the altermagnet (Han et al., 2024). First-principles calculations for Pt give magnetic-octupole Hall conductivities

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$64

comparable to the conventional spin Hall conductivity of Pt (Han et al., 2024). A complementary theory shows that the magnetic octupole Hall effect in d-wave altermagnets persists even in symmetries where the spin-splitter effect is forbidden, and that a sizable electric quadrupole Hall effect is also symmetry-allowed (Ko et al., 1 Aug 2025).

Strain provides another control parameter. Symmetry analysis over collinear spin point groups identifies 15 SPGs that admit strain-induced nonrelativistic Zeeman-type spin splittings,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$65

and these 15 coincide exactly with the cases associated with d-wave altermagnetic spin splittings in the literature (Zhai et al., 9 Jun 2025). First-principles calculations show that a shear strain of $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$66 creates nonrelativistic ZSSs of $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$67 in CoF$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$68, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$69 in LiFe$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$70F$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$71, and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$72 in La$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$73O$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$74Mn$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$75Se$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$76 (Zhai et al., 9 Jun 2025).

Collective excitations also inherit the d-wave structure. In a continuum electron model with

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$77

RPA yields a spin demon: an acoustic, electrically neutral spin-plasmon consisting of out-of-phase oscillations of the two spin species (Gunnink et al., 15 Apr 2025). The mode lives outside the particle-hole continuum of one spin species, reaches quality factors $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$78, carries a magnetic moment, and changes sign under a $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$79 rotation, which is the hallmark of the underlying d-wave order (Gunnink et al., 15 Apr 2025).

6. Topology, intertwined phases, and unresolved material questions

Two-dimensional d-wave altermagnets support topological phase transitions, edge-state physics, and spin-selective real-space responses. In the tight-binding model with critical intra-sublattice hopping $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$80, the system changes from a gapped band insulator to a topological nodal semimetal with Dirac points, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$81 Berry flux per node, conductivity anisotropy, spin-dependent “steering” effects, and edge-state markers based on fidelity susceptibility and inverse participation ratio (Calixto, 4 Feb 2026). In ultranarrow ribbons, hybridization of edge states opens a controllable energy gap,

$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$82

which is used to propose a topological altermagnetic field-effect transistor (Calixto, 4 Feb 2026).

The same d-wave spin splitting can stabilize additional ordered phases. A non-perturbative static path approximation Monte Carlo study shows that a two-dimensional d-wave altermagnet supports a robust pair-density-wave phase over a finite temperature window, with distinct thermal scales $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$83, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$84, and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$85 (Madhusuthanan et al., 8 May 2026). In a different strongly correlated setting, constrained-path quantum Monte Carlo on a Hubbard model with spin-anisotropic hopping finds that increasing anisotropy suppresses long-range antiferromagnetic order and significantly enhances effective $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$86-wave pairing correlations, providing a doping-free route to unconventional superconductivity mediated by short-range spin fluctuations in an altermagnetic background (Li et al., 18 May 2025).

Other intertwined responses include d-wave polarization–spin locking in tetragonal two-dimensional altermagnets, where spin-up and spin-down electrons accumulate at orthogonal edges; monolayer Cr$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$87X$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$88O ($\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$89) is proposed as a candidate, with $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$90 and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$91 in lattice units (Liu et al., 22 Feb 2025). Magnon–phonon hybridization in a square-lattice d-wave altermagnet yields magnon polarons with finite phonon angular momentum, and the phonon angular-momentum texture follows the same d-wave form factor as the magnon spin texture (Bendin et al., 11 Nov 2025). A multi-field proposal further combines gating, circularly polarized light, and in-plane electric fields to generate tunable $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$92-, $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$93-, and $\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$94-spin polarizations in two-dimensional d-wave altermagnets (Yarmohammadi et al., 2 Oct 2025).

One material-specific issue remains explicit in the literature. RuO$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$95 is used as a prototypical d-wave altermagnet in optical calculations and thin-film magneto-optical experiments (Weber et al., 2024), while a separate comparison states that for RuO$\Delta_d(\mathbf k)=\Delta_0(\cos k_x-\cos k_y)$96 there are “controversies over whether true collinear altermagnetism exists; any splitting tends to be small” (Jiang et al., 2024). The coexistence of such statements indicates that, for at least some candidate systems, symmetry-based theory, thin-film measurements, and microscopic magnetic structure remain an active point of comparison rather than a universally settled classification.