---
title: 'P-Wave Altermagnets: Symmetry & Transport'
url: https://www.emergentmind.com/topics/p-wave-altermagnets
type: topic
---

# P-Wave Altermagnets: Symmetry & Transport

P-wave altermagnets occupy a non-uniform but increasingly important place in the current literature on unconventional magnetism. One strand classifies altermagnets by angular-momentum harmonics in momentum space, \(\boldsymbol{\sigma}\,Y_\ell^m(\theta,\phi)\,|{\bf k}|^\ell\), so that \(\ell=1,2,3\) correspond to p-, d-, and f-wave altermagnets [2403.14620]. Another strand distinguishes even-parity altermagnets from odd-parity altermagnets, with the odd-parity class obeying \(E(\mathbf{k},s)=E(-\mathbf{k},-s)\) and explicitly including odd parity p-wave altermagnets [2508.18360]. A third strand reserves “altermagnets” for collinear even-parity nodal magnets and treats p-wave magnets as odd-parity, non-collinear analogs beyond the strict altermagnetic definition [2409.10034]. Across these usages, the recurring motif is compensated magnetism with momentum-dependent spin splitting whose leading reciprocal-space structure is p-wave.

## 1. Terminology and classification

Altermagnets have been defined as “crystallographic rotational symmetry breaking spin-ordered states, possessing a net zero magnetization despite manifesting Kramers non-degenerate bands” [2403.14620]. In that framework, the band splitting is classified by angular harmonics in reciprocal space, and monolayer, Bernal bilayer, and rhombohedral trilayer graphene respectively realize p-, d-, and f-wave altermagnets when momentum-independent local spin nematic orders are projected into the band basis [2403.14620].

A different but compatible classification is based on the behavior of spin splitting at time-reversal-related momenta. Even-parity altermagnets satisfy
\[
E(\mathbf{k},s)=E(-\mathbf{k},s),
\]
whereas odd-parity altermagnets satisfy
\[
E(\mathbf{k},s)=E(-\mathbf{k},-s).
\]
Within this language, circularly polarized light can dynamically convert collinear PT-symmetric antiferromagnets on dimerized lattices into odd parity p-wave altermagnets [2508.18360].

The terminology becomes less uniform in the spin-group literature. The review of nodal magnetically ordered phases states that collinear altermagnets realize even-parity nodal order, while p-wave magnets are the odd-parity, non-collinear analogs with symmetry \([C_2\parallel \mathbf{t}]\), zero net magnetization, and parity-breaking spin-polarized Fermi surfaces that shift in opposite directions in momentum space for opposite spin directions [2409.10034]. The ferroelectric literature adds the labels “time-reversal-symmetric p-wave magnets” and “antialtermagnets” for odd-parity-wave magnets with noncollinear magnetic sublattices and time-reversal-symmetric momentum-space spin polarization [2603.19107]. This suggests that “p-wave altermagnet” currently functions as a broad umbrella for several closely related symmetry settings rather than a single universally fixed definition.

## 2. Symmetry structure and reciprocal-space geometry

In the graphene construction, the free Hamiltonian near a valley is
\[
\hat{h}_1({\bf k}) = \alpha_1 |{\bf k}| \left[ \Gamma_1^{1}\cos\phi - \Gamma_1^{2}\sin\phi \right],
\]
and local spin nematic order, once projected into the band basis, acquires angular dependence through \(\cos(\ell\phi)\) and \(\sin(\ell\phi)\). For \(\ell=1\), the intraband term is proportional to \(\cos\phi\) and \(\sin\phi\), which are the basic p-wave harmonics and can be written in terms of \(Y_1^m\) spherical harmonics [2403.14620]. In monolayer graphene, the constant-energy contours of opposite spins are non-overlapping except at discrete crossing points, but the enclosed areas remain equal, so the net magnetization vanishes despite spin splitting [2403.14620].

The non-collinear p-wave-magnet formulation emphasizes a different symmetry pattern. There the defining band relation is
\[
E_{\sigma}(\mathbf{k}) = E_{-\sigma}(-\mathbf{k}),
\]
with parity-breaking spin-polarized Fermi surfaces shifted in opposite directions for opposite spin projections [2409.10034]. The same time-reversal-symmetric momentum-space relation is written in the superconductivity context as
\[
E_{\mathbf{k}\uparrow}=E_{-\mathbf{k}\downarrow},
\]
together with the statements that p-wave magnets have zero net magnetization by symmetry, finite non-relativistic spin splitting of electron bands, and spin polarizations collinear in momentum space, while the real-space magnetization is noncollinear [2601.19829].

The relativistic extension shows that odd and even wave symmetries can coexist in different spin components. In centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe, the dominant spin component retains \(g\)-wave character in the relativistic regime only when the Néel vector is oriented along the \(z\)-axis, while subdominant components exhibit \(d\)-wave symmetry in CrSb and \(p\)-wave symmetry in ferroelectric wurtzite MnTe [2605.23438]. More generally, the \(g\)-wave character survives in the relativistic limit only when both the Néel vector and the electric field associated with inversion-symmetry breaking are oriented along \(z\) [2605.23438].

## 3. Microscopic constructions

A first microscopic route starts from momentum-independent local spin nematicity in graphene systems. Monolayer graphene realizes the p-wave case, Bernal bilayer the d-wave case, and rhombohedral trilayer the f-wave case, with the angular momentum \(\ell\) inherited from the linear, quadratic, and cubic free-fermion dispersions [2403.14620]. The p-wave case is therefore tied directly to Dirac band topology and to the projection of a local order parameter into a band basis with \(\ell=1\) harmonics.

A second route is Floquet engineering on dimerized lattices. For a 2D dimerized square lattice with collinear antiferromagnetic order, the static Hamiltonian is
\[
\mathcal{H}^{\rm (I)}(\mathbf{k})=
(t_+\cos k_{x}+2t_{2}\cos k_{y})\sigma_{x}
+t_{-}\sin k_{x}\sigma_{y}
+\mathbf{M}\cdot\mathbf{s}\,\sigma_z.
\]
Under circularly polarized light in the high-frequency off-resonant regime, the effective Floquet Hamiltonian becomes
\[
\mathcal{H}_{\rm eff}(\mathbf{k})=
J_0(A_0)\big[(t_+\cos k_{x}+2t_{2}\cos k_{y})\sigma_{x}+t_-\sin k_{x}\sigma_{y}\big]
+M\sigma_{z}s_{z}
-\eta F(A_{0},\omega)\sin k_{y}\cos k_{x}\sigma_{z},
\]
and the new \(\sin k_y \cos k_x\) term generates odd-parity momentum-dependent spin splitting. In this setting, circularly polarized light dynamically converts the collinear PT-symmetric antiferromagnet into an odd parity p-wave altermagnet [2508.18360].

A third route uses continuum and tight-binding models for p-wave magnets. The phenomenological continuum Hamiltonian
\[
H(\mathbf{k}) = \frac{\hbar^2}{2m}\left[ \left(k^2 + \boldsymbol{\alpha}^2 \right)\sigma_0 + 2\,\mathbf{k}\cdot\boldsymbol{\alpha}\,\sigma_z \right] - \mu
\]
shifts the spin-up and spin-down bands in momentum space by \(\pm\boldsymbol{\alpha}\), while the square-lattice realization
\[
\mathcal{H}(\mathbf{k}) = -t_0\big(\cos k_x a + \cos k_y a\big)\sigma_0 + \big(t_x \sin k_x a + t_y \sin k_y a\big)\sigma_z
\]
implements the same symmetry at the lattice level [2408.10413]. These models satisfy
\[
E_\sigma(\mathbf{k}) \neq E_\sigma(-\mathbf{k}),\qquad E_\sigma(\mathbf{k}) = E_{-\sigma}(-\mathbf{k}),
\]
which is the hallmark odd-parity relation used in the p-wave-magnet literature [2408.10413].

## 4. Transport and response phenomenology

One of the sharpest distinctions between p-wave and higher-wave altermagnets appears in thermal spin transport. Within the \(X\)-wave magnet framework \(X=s,p,d,f,g,i\), the Boltzmann expansion in powers of \(\nabla T\) yields a hierarchy of spin-Seebeck and spin-Nernst responses, but “no linear nor nonlinear spin current is generated in \(p\)-wave magnets” [2602.19034]. In the notation of that work,
\[
j_{\text{spin}}^{(x;x)}=j_{\text{spin}}^{(x^2;x)}=j_{\text{spin}}^{(x^3;x)}=\cdots =0,
\qquad
j_{\text{spin}}^{(x;y)}=j_{\text{spin}}^{(x^2;y)}=j_{\text{spin}}^{(x^3;y)}=\cdots =0.
\]
This null result is exact within the nonrelativistic exchange model and contrasts with the linear spin-Nernst response of d-wave altermagnets, the second-order spin-Seebeck diode of f-wave magnets, the third-order spin-Nernst response of g-wave altermagnets, and the linear transverse spin-Nernst response of i-wave altermagnets [2602.19034].

In spin-orbit-coupled backgrounds, p-wave order modifies mixed magnetic responses in a more conventional way. For Rashba metals, the p-wave order parameter exerts only a limited influence on the orbital-Zeeman cross term, whereas on the surface of a three-dimensional topological insulator the p-wave order retains the step-function-type dependence of the orbital-Zeeman term as a function of \(\mu\), associated with the jump at \(\mu=0\), but reduces its magnitude [2603.09325]. At \(T=0\),
\[
\chi_{\rm OZ}^{(p)}
=
\mathrm{sgn}(\mu)\,
\frac{|e|\mu_{\rm B}}{2\pi\hbar}\,
\frac{\lambda}{\sqrt{\lambda^2 + J^2}}.
\]

Junction transport provides a different signature. For a normal metal/p-wave magnet junction in the ballistic regime, the mirror relation
\[
T_\uparrow(E,\theta)=T_\downarrow(E,-\theta)
\]
implies zero transverse charge current but finite transverse spin conductance,
\[
\frac{G_s(E)}{G_s^0} = \int_{-\pi/2}^{\pi/2} \big[ T_\uparrow(E,\theta)\sin\theta'_\uparrow - T_\downarrow(E,\theta)\sin\theta'_\downarrow \big]\, d\theta,
\]
so that a pure transverse spin current flows parallel to the interface [2408.10413]. The same model exhibits an indirect conductance gap for \(E<\Delta=\alpha_y^2/4\) when the normal and p-wave Fermi circles no longer overlap [2408.10413].

## 5. Topological and superconducting extensions

Floquet odd-parity p-wave altermagnets on dimerized lattices realize topological electronic phases because the underlying Dirac structure allows mass inversion under drive. In 2D, the light-induced odd-parity p-wave altermagnet becomes a Chern insulator with total Chern number
\[
C=2\eta,
\]
while in 3D the corresponding driven system becomes a Weyl semimetal with spin-resolved Weyl nodes and Fermi-arc surface states [2508.18360]. The same work emphasizes that the direction of spin splitting is perpendicular to the dimerization direction and can be controlled by the drive geometry [2508.18360].

Relativistic ferroelectric altermagnets add a further p-wave route. In wurtzite MnTe, selected bands exhibit p-wave magnetism when the Néel vector is aligned along the \(x\)-axis, and the relativistic spin-momentum locking can show one symmetry-protected nodal plane and an accidental nodal surface, or two accidental nodal surfaces, depending on the spin component and the odd-even mixture of dipole and quadrupole terms [2605.23438]. The important point is not a complete replacement of even-wave altermagnetism, but a band- and spin-component-resolved coexistence of \(p\)-, \(d\)-, and \(g\)-wave characters [2605.23438].

Superconducting extensions split into two distinct problems. In p-wave magnets, if superconductivity develops, “the only supported superconducting symmetry is Ising superconductivity,” and “any Cooper pair is a 50:50 mix of singlet and triplet” [2601.19829]. This is tied to the relation \(E_{\mathbf{k}\uparrow}=E_{-\mathbf{k}\downarrow}\) and to the large non-relativistic spin splitting [2601.19829]. Separately, in three-dimensional \(g\)-wave altermagnetic metals, the altermagnetic spin splitting can stabilize chiral \(p\)-wave superconducting states under strong altermagnetic fields and high electron densities [2602.22736]. In that context, “p-wave” refers to the pairing channel rather than to the magnetic symmetry itself, and the distinction is essential.

The graphene-based spin-nematic framework also has a superconducting analogue: the same band-topology logic leads to p-wave, d-wave, and f-wave Majorana altermagnets in spin-triplet nematic superconductors built on monolayer, bilayer, and trilayer graphene, respectively [2403.14620].

## 6. Materials, probes, and unresolved nomenclature

The present candidate set is diverse. Monolayer graphene is the canonical continuum p-wave altermagnet in the local-spin-nematic construction [2403.14620]. CeNiAsO is highlighted as a predicted p-wave magnet in the spin-group review [2409.10034]. Mn\(_5\)Si\(_3\) hosts two possible classes of unconventional p-wave magnetism, coplanar and non-coplanar, with only the non-coplanar configuration surviving the spin-symmetry requirements for nonlinear shift photocurrent [2406.19842]. Ferroelectric odd-parity-wave magnets add a materials roadmap: the classification based on crystallographic, exchange-driven, and spin-orbit-driven polar symmetry breaking yields more than 50 candidate materials, and first-principles calculations identify a pristine, time-reversal-symmetric p-wave spin-polarized electronic structure in \(\mathrm{GdMn_2O_5}\), with electrically switchable p-wave order [2603.19107]. Wurtzite MnTe enters as the ferroelectric relativistic case with band-selective p-wave magnetism [2605.23438].

The experimental toolbox is correspondingly broad. Spin-resolved ARPES is proposed for graphene-based p-wave altermagnets because it can reveal spin-split contours with dipolar angular dependence and zero integrated magnetization [2403.14620]. Photogalvanic probes are symmetry-selective in weak-SOC p-wave magnets: in Mn\(_5\)Si\(_3\), the dominant shift-current tensor components are determined by the spin group rather than the magnetic point group, providing a direct route to distinguish coplanar from non-coplanar p-wave order [2406.19842]. Tunneling probes access impurity responses: in unconventional p-wave magnets, the impurity-induced LDOS is anisotropic, and near the impurity the LDOS can show oscillations with a doubled period originating from the interplay of propagating and evanescent waves [2408.01494]. Junction conductance, orbital magnetization jumps on topological-insulator surfaces, and electrically driven polarization switching in ferroelectric p-wave magnets complete the current set of proposed diagnostics [2408.10413, 2603.09325, 2603.19107].

A recurring misconception is that all p-wave spin-split compensated magnets are simply another altermagnetic subtype. The literature itself is more divided. Some works explicitly use “p-wave altermagnets” for odd-parity or band-projected \(\ell=1\) cases [2403.14620, 2508.18360], whereas others treat p-wave magnets as odd-parity, non-collinear analogs beyond strict altermagnetism [2409.10034], or as time-reversal-symmetric p-wave magnets and antialtermagnets [2603.19107]. A cautious synthesis is therefore that p-wave altermagnets are best understood as the \(\ell=1\), odd-parity frontier of compensated spin-split magnetism, with the precise symmetry definition depending on whether the starting point is band-projected spin nematicity, odd-parity Floquet altermagnetism, spin-group p-wave magnetism, or relativistic mixed odd-even spin-momentum locking.

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