---
title: D-Wave Altermagnets
url: https://www.emergentmind.com/topics/d-wave-altermagnets
type: topic
---

# D-Wave Altermagnets

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 $\lambda (k_x^2-k_y^2)$ [2408.05187, 2509.08254]. 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 [2408.05187]. 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 [2512.24114, 2412.13763].

## 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 [2512.24114]. 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 $C_4\Theta$ [2512.24114, 2410.14542]. 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 $C_4^z$, $f_d$ changes sign, while under $C_2^z$ it is invariant; this sign reversal encodes the d-wave character of the altermagnetic order [2512.24114]. In La$_2$O$_3$Mn$_2$Se$_2$, symmetry analysis in $D_{4h}$ assigns the altermagnetic order parameter to the $B_2g$ irreducible representation with simplest basis function $\psi_d(\mathbf k)\propto k_x^2-k_y^2$ [2410.14542]. In CsV$_2$Se$_2$O, the same sign-changing structure is described as a $B_{1g}$ form factor $f_d(\mathbf k)=\cos k_x-\cos k_y$ [2512.24114]. This suggests that the essential invariant across notational conventions is the alternating sign structure under $C_4$ rather than a single universal label.

Several works formulate the d-wave order through a spin-resolved expectation value. One compact expression is
$$
\eta_d\equiv \langle S_z f_d(\mathbf k)\rangle,
$$
with $f_d(\mathbf k)=\cos k_x-\cos k_y$ [2512.24114]. Another is the momentum-space spin density constraint
$$
S_z(\mathbf k)-S_z(-\mathbf k)\propto \psi_d(\mathbf k),
$$
which makes explicit that the net magnetization still vanishes after Brillouin-zone integration [2410.14542]. 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,
$$
H(\mathbf k)=\varepsilon(\mathbf k)\sigma_0+\Delta(\mathbf k)\sigma_z,
$$
with $\Delta(\mathbf k)\propto k_x^2-k_y^2$ or $\cos k_x-\cos k_y$ [2410.14542, 2408.00320]. For La$_2$O$_3$Mn$_2$Se$_2$, one $k\cdot p$ form is
$$
\varepsilon(\mathbf k)=\varepsilon_0+\alpha(k_x^2+k_y^2),\qquad
\Delta(\mathbf k)=\lambda(k_x^2-k_y^2),
$$
giving
$$
E_\pm(\mathbf k)=\varepsilon(\mathbf k)\pm |\Delta(\mathbf k)|.
$$
The spin splitting vanishes along the nodal planes $k_x=\pm k_y$ and reaches extrema along the principal axes [2410.14542].

A continuum model used for nonlinear opto-magnetic response is
$$
H(\mathbf k)=J(k_x^2-k_y^2)\sigma_z,
$$
whose two bands $\varepsilon_\pm(\mathbf k)=\pm J(k_x^2-k_y^2)$ are fully spin-split, yet the net magnetization
$$
\int_{\mathrm{BZ}}d^2k\,[f(\varepsilon_+)-f(\varepsilon_-)]=0
$$
remains zero [2509.08254]. A lattice realization for KRu$_4$O$_8$ is
$$
H(\mathbf k)=h_0(\mathbf k)+h_a(\mathbf k)\,\hat n\cdot \sigma,
$$
with
$$
h_0=-t(\cos k_x+\cos k_y),\qquad
h_a=-t_J(\cos k_x-\cos k_y)\propto \Delta(\mathbf k),
$$
and parameters $t=0.1\,\mathrm{eV}$, $t_J=0.075\,\mathrm{eV}$, $\mu=-0.1\,\mathrm{eV}$ [2509.08254].

For metallic KV$_2$Se$_2$O, the band structure is modeled as
$$
H(\mathbf k)=\varepsilon(\mathbf k)\,\mathbb 1_2+\lambda\,\Delta(\mathbf k)\sigma_z,
$$
with
$$
\varepsilon(\mathbf k)=-2t[\cos(k_xa)+\cos(k_ya)]-4t'\cos(k_xa)\cos(k_ya)-\mu,
$$
$$
\Delta(\mathbf k)=\Delta_0[\cos(k_xa)-\cos(k_ya)].
$$
The eigenvalues
$$
E_\pm(\mathbf k)=\varepsilon(\mathbf k)\pm \lambda \Delta(\mathbf k)
$$
yield spin-split bands along $\Gamma$–X and degeneracy along the nodal lines $k_x=\pm k_y$ [2408.00320].

A separate tight-binding formulation emphasizes topological structure in two dimensions:
$$
H_s(\mathbf k)=d_x^s(\mathbf k)\tau_x+d_y^s(\mathbf k)\tau_y+d_z^s(\mathbf k)\tau_z,
$$
with
$$
d_z^s(\mathbf k)=sJ+2t_a(\cos k_x-\cos k_y).
$$
In that model, the direct gap closes at the critical point
$$
t_a^C=J/4,
$$
and for $t_a>t_a^C$ the system enters a topological nodal semimetal with Dirac points, Berry-curvature singularities, and pseudospin-texture winding [2602.04854].

## 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$_2$O$_3$Mn$_2$Se$_2$ | correlated insulating layered d-wave altermagnet | $T_N\simeq 166\,\mathrm K$; full insulating gap $>1.5\,\mathrm{eV}$; spin splitting up to $\sim 0.7\,\mathrm{eV}$; robust 2D AFM fluctuations above $T_N$ [2410.14542] |
| KV$_2$Se$_2$O | metallic room-temperature d-wave altermagnet | highly anisotropic spin-polarized Fermi surfaces; SDW below $T_{\mathrm{SDW}}\approx 100\,\mathrm K$ [2408.00320, 2505.00074] |
| CsV$_2$Se$_2$O | candidate material with atomic-scale visualization | unidirectional electronic patterns tied to magnetic domain walls and spin defects; elliptical charging rings; SDW gap $\Delta_{\mathrm{SDW}}\approx 70\,\mathrm{meV}$ [2512.24114] |
| RbV$_2$Se$_2$O | 2D d-wave altermagnet with spin-texture locking | spin-lattice, spin-scattering, spin-momentum, and spin-stripe locking visualized by SP-STM and QPI [2604.18337] |
| RuO$_2$ | prototypical d-wave altermagnet in optical studies | optical analogue of a spin splitter effect; pump-polarization-dependent persistent optically excited electronic spin polarization [2408.05187] |

In La$_2$O$_3$Mn$_2$Se$_2$, the parent crystal is body-centered tetragonal $I4/mmm$, neutron diffraction finds G-type antiferromagnetic order at $\mathbf k=(0,0,0)$, and the resulting magnetic space group is $I4'/mm'm$ [2410.14542]. DFT+$U$ with $U_{\mathrm eff}\approx 4\,\mathrm{eV}$ on Mn-$d$ predicts a full insulating gap larger than $1.5\,\mathrm{eV}$ and momentum-dependent spin splitting up to $\sim 0.7\,\mathrm{eV}$ along $\Gamma$–X, reversing sign along $\Gamma$–Y, with nodal planes at $k_x=\pm k_y$ [2410.14542]. Magnetometry shows a sharp kink at $T_N\simeq 166\,\mathrm K$, while neutron pair distribution function analysis shows a 2D short-range magnetic component that persists above the Néel temperature [2410.14542].

KV$_2$Se$_2$O is reported as a metallic room-temperature d-wave altermagnet with crystal space group $P4/mmm$ and a zero-temperature collinear order above $100\,\mathrm K$ described by the spin-space symmetry operation $[C_2|C_{4z}]$ [2408.00320]. Below $T_{\mathrm{SDW}}\approx 100\,\mathrm K$, 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 [2505.00074].

Direct real-space evidence is reported for CsV$_2$Se$_2$O and RbV$_2$Se$_2$O. In CsV$_2$Se$_2$O, STM at $4\,\mathrm K$ resolves strip-like domain walls along $(110)$, defect-bound in-gap resonances at $\sim 20\,\mathrm{meV}$ and $\sim 40\,\mathrm{meV}$, quasi-1D charge modulation of wavelength $2a_0$, and elliptical charging rings elongated along either $x$ or $y$ [2512.24114]. In RbV$_2$Se$_2$O, SP-STM with a field-switchable Cr tip at $T\approx 1.7\,\mathrm K$ reveals sublattice-resolved out-of-plane spin contrast, spin-dependent QPI anisotropy, and a long-period stripe modulation with $\lambda_{\mathrm{stripe}}\simeq 6\,\mathrm{nm}$, interpreted as quadruple spin-texture locking [2604.18337].

## 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,
$$
M_i=\chi_{ijl}(\Omega)\,E_j(\Omega)E_l(-\Omega).
$$
For a d-wave altermagnet with combined spin-inversion $C_2$ and fourfold rotation $C_{4z}$ symmetry,
$$
\chi_{ixx}=-\chi_{iyy},\qquad \chi_{ixy}=\chi_{iyx},
$$
and for in-plane polarization angle $\phi$,
$$
M_i=[\chi_{ixx}\cos 2\phi+\chi_{ixy}\sin 2\phi]E^2.
$$
When a mirror plane makes $\chi_{ixy}=0$, the result reduces to
$$
M=\hat n\,\chi\,E^2\cos 2\phi,
$$
so the induced magnetization is locked to the Néel vector and has $\pi$ periodicity in the pump polarization angle [2509.08254].

For the KRu$_4$O$_8$ parameter set, numerical evaluation gives a dip in $\chi/\chi_0$ at terahertz frequencies and a sign change at higher $\Omega$, with $\chi_0=e^2\mu_B/t^2$ [2509.08254]. Choosing $E\approx 1\,\mathrm{MV\,cm^{-1}}$ gives $\chi_0E^2\sim \mu_B\,\mathrm{nm}^2$, and for $\Omega=10\,\mathrm{THz}$, $\Gamma=0.01\,\mathrm{eV}$, and $T=100\,\mathrm K$ one finds $\chi/\chi_0\approx -0.016$, implying
$$
M\approx -0.016\,\mu_B\ \text{per unit cell}.
$$
For $d_{x^2-y^2}$ order, $M(\phi)\propto \cos 2\phi$ with maxima at $\phi=0,\pi$ and zero crossings at $\phi=\pi/4,3\pi/4$; for $d_{xy}$ order, $M\propto \sin 2\phi$ [2509.08254].

An experimentally realized optical analogue of the spin-splitter effect was reported for ultrathin RuO$_2$ films. In the ab-initio description, the optically excited spin polarization obeys
$$
S^z_{\mathrm{exc}}(\phi)\propto A(\hbar\omega_L)\sin(2\phi),
$$
with zero crossings at $\phi=0^\circ,90^\circ,180^\circ$ [2408.05187]. Pump–probe measurements on a $3.5\,\mathrm{nm}$ epitaxial RuO$_2$(001) film used a $40\,\mathrm{fs}$, $1.5\,\mathrm{eV}$ pump and a $3.1\,\mathrm{eV}$ polar-MOKE probe. At $t\approx 50\,\mathrm{fs}$, the Kerr signal showed a clear $\sin(2\phi)$ periodicity with a maximum at $\phi=45^\circ$ and opposite sign at $\phi=135^\circ$; by $t\approx 5\,\mathrm{ps}$, the $180^\circ$ modulation had vanished [2408.05187].

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 [2604.12726]. In the ideal limit $V_\delta=0$, closed-form analytical expressions are obtained; for finite $V_\delta\ll V_\pi$, perturbative analytical results agree well with numerical calculations up to $V_\delta/V_\pi\approx 0.3$ [2604.12726]. In the clean limit, the third-order injection current exceeds the shift current by $\sim 10^4$, and in the ideal limit the sublattice–orbital–spin locking yields $100\%$ spin-polarized third-order currents [2604.12726].

## 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 [2412.13763]. 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 [2412.13763]. The same framework yields a distinctive spin-transfer torque,
$$
\bm\tau_{\bm m,\mathrm{ST}}^{\mathrm{AM}}
=
\eta_{\mathrm{FL}}[j_x^e\partial_y+j_y^e\partial_x]\bm n
+\frac{\eta_{\mathrm{DL}}}{\mathfrak s}\bm n\times [j_x^e\partial_y+j_y^e\partial_x]\bm n,
$$
and predicts domain-wall motion driven by transverse charge currents [2412.13763].

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 [2409.14423]. First-principles calculations for Pt give magnetic-octupole Hall conductivities
$$
\chi_{zx}^{O^x_{xy}}\simeq -1303\ (\hbar/e)(\Omega\cdot\mathrm{cm})^{-1},\qquad
\chi_{zx}^{O^z_{yz}}\simeq +1569\ (\hbar/e)(\Omega\cdot\mathrm{cm})^{-1},
$$
comparable to the conventional spin Hall conductivity of Pt [2409.14423]. 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 [2508.00794].

Strain provides another control parameter. Symmetry analysis over collinear spin point groups identifies 15 SPGs that admit strain-induced nonrelativistic Zeeman-type spin splittings,
$$
\tilde H_{\mathrm eff}(\{\eta_{\alpha\beta}\})=\sum_{\alpha\beta}\tilde\lambda_{\alpha\beta}\eta_{\alpha\beta}\sigma_\chi,
$$
and these 15 coincide exactly with the cases associated with d-wave altermagnetic spin splittings in the literature [2506.07447]. First-principles calculations show that a shear strain of $2\%$ creates nonrelativistic ZSSs of $177\,\mathrm{meV}$ in CoF$_2$, $100\,\mathrm{meV}$ in LiFe$_2$F$_6$, and $102\,\mathrm{meV}$ in La$_2$O$_3$Mn$_2$Se$_2$ [2506.07447].

Collective excitations also inherit the d-wave structure. In a continuum electron model with
$$
\varepsilon_{\mathbf k}^\sigma=\frac{\hbar^2}{2m_0}|\mathbf k|^2+\sigma\frac{\hbar^2}{2m_*}(k_x^2-k_y^2),
$$
RPA yields a spin demon: an acoustic, electrically neutral spin-plasmon consisting of out-of-phase oscillations of the two spin species [2504.11062]. The mode lives outside the particle-hole continuum of one spin species, reaches quality factors $>10$, carries a magnetic moment, and changes sign under a $90^\circ$ rotation, which is the hallmark of the underlying d-wave order [2504.11062].

## 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 $t_a^C=J/4$, the system changes from a gapped band insulator to a topological nodal semimetal with Dirac points, $\pi$ Berry flux per node, conductivity anisotropy, spin-dependent “steering” effects, and edge-state markers based on fidelity susceptibility and inverse participation ratio [2602.04854]. In ultranarrow ribbons, hybridization of edge states opens a controllable energy gap,
$$
\Delta(L_y)=\Delta_0\exp[-L_y/\lambda+\xi/L_y],
$$
which is used to propose a topological altermagnetic field-effect transistor [2602.04854].

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 $T_c$, $T_{\mathrm gap}$, and $T_{pg}$ [2605.07656]. 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 $d$-wave pairing correlations, providing a doping-free route to unconventional superconductivity mediated by short-range spin fluctuations in an altermagnetic background [2505.12342].

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$_2$X$_2$O ($X=\mathrm{Se},\mathrm{Te}$) is proposed as a candidate, with $\vec p_\uparrow=(0,\tfrac12)$ and $\vec p_\downarrow=(\tfrac12,0)$ in lattice units [2502.16103]. 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 [2511.08357]. A multi-field proposal further combines gating, circularly polarized light, and in-plane electric fields to generate tunable $x$-, $y$-, and $z$-spin polarizations in two-dimensional d-wave altermagnets [2510.02452].

One material-specific issue remains explicit in the literature. RuO$_2$ is used as a prototypical d-wave altermagnet in optical calculations and thin-film magneto-optical experiments [2408.05187], while a separate comparison states that for RuO$_2$ there are “controversies over whether true collinear altermagnetism exists; any splitting tends to be small” [2408.00320]. 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.

Source: https://www.emergentmind.com/topics/d-wave-altermagnets