P-Wave Altermagnets: Symmetry & Transport

• P-wave altermagnets are unconventional, compensated spin-split magnets characterized by momentum-space p-wave harmonics and odd-parity symmetry.
• Experimental probes like spin-resolved ARPES and tunneling reveal anisotropic spin textures and distinct transport signatures.
• Theoretical models, including graphene-based spin nematicity and Floquet engineering, elucidate their unique magnetic, topological, and superconducting properties.

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 (Das et al., 2024). 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 (Liu et al., 25 Aug 2025). 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 (Jungwirth et al., 2024). 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” (Das et al., 2024). 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 (Das et al., 2024).

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 (Liu et al., 25 Aug 2025).

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 (Jungwirth et al., 2024). 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 (Priessnitz et al., 19 Mar 2026). 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 $\ell=1,2,3$0 and $\ell=1,2,3$1, which are the basic p-wave harmonics and can be written in terms of $\ell=1,2,3$2 spherical harmonics (Das et al., 2024). 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 (Das et al., 2024).

The non-collinear p-wave-magnet formulation emphasizes a different symmetry pattern. There the defining band relation is

$\ell=1,2,3$3

with parity-breaking spin-polarized Fermi surfaces shifted in opposite directions for opposite spin projections (Jungwirth et al., 2024). The same time-reversal-symmetric momentum-space relation is written in the superconductivity context as

$\ell=1,2,3$4

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 (Khodas et al., 27 Jan 2026).

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 $\ell=1,2,3$5-wave character in the relativistic regime only when the Néel vector is oriented along the $\ell=1,2,3$6-axis, while subdominant components exhibit $\ell=1,2,3$7-wave symmetry in CrSb and $\ell=1,2,3$8-wave symmetry in ferroelectric wurtzite MnTe (Gong et al., 22 May 2026). More generally, the $\ell=1,2,3$9-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 $E(\mathbf{k},s)=E(-\mathbf{k},-s)$0 (Gong et al., 22 May 2026).

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 $E(\mathbf{k},s)=E(-\mathbf{k},-s)$1 inherited from the linear, quadratic, and cubic free-fermion dispersions (Das et al., 2024). 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 $E(\mathbf{k},s)=E(-\mathbf{k},-s)$2 harmonics.

A second route is Floquet engineering on dimerized lattices. For a 2D dimerized square lattice with collinear antiferromagnetic order, the static Hamiltonian is

$E(\mathbf{k},s)=E(-\mathbf{k},-s)$3

Under circularly polarized light in the high-frequency off-resonant regime, the effective Floquet Hamiltonian becomes

$E(\mathbf{k},s)=E(-\mathbf{k},-s)$4

and the new $E(\mathbf{k},s)=E(-\mathbf{k},-s)$5 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 (Liu et al., 25 Aug 2025).

A third route uses continuum and tight-binding models for p-wave magnets. The phenomenological continuum Hamiltonian

$E(\mathbf{k},s)=E(-\mathbf{k},-s)$6

shifts the spin-up and spin-down bands in momentum space by $E(\mathbf{k},s)=E(-\mathbf{k},-s)$7, while the square-lattice realization

$E(\mathbf{k},s)=E(-\mathbf{k},-s)$8

implements the same symmetry at the lattice level (Salehi et al., 2024). These models satisfy

$E(\mathbf{k},s)=E(-\mathbf{k},-s)$9

which is the hallmark odd-parity relation used in the p-wave-magnet literature (Salehi et al., 2024).

4. Transport and response phenomenology

One of the sharpest distinctions between p-wave and higher-wave altermagnets appears in thermal spin transport. Within the $E(\mathbf{k},s)=E(-\mathbf{k},s),$0-wave magnet framework $E(\mathbf{k},s)=E(-\mathbf{k},s),$1, the Boltzmann expansion in powers of $E(\mathbf{k},s)=E(-\mathbf{k},s),$2 yields a hierarchy of spin-Seebeck and spin-Nernst responses, but “no linear nor nonlinear spin current is generated in $E(\mathbf{k},s)=E(-\mathbf{k},s),$3-wave magnets” (Ezawa, 22 Feb 2026). In the notation of that work,

$E(\mathbf{k},s)=E(-\mathbf{k},s),$4

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 (Ezawa, 22 Feb 2026).

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 $E(\mathbf{k},s)=E(-\mathbf{k},s),$5, associated with the jump at $E(\mathbf{k},s)=E(-\mathbf{k},s),$6, but reduces its magnitude (Mizoguchi et al., 10 Mar 2026). At $E(\mathbf{k},s)=E(-\mathbf{k},s),$7,

$E(\mathbf{k},s)=E(-\mathbf{k},s),$8

Junction transport provides a different signature. For a normal metal/p-wave magnet junction in the ballistic regime, the mirror relation

$E(\mathbf{k},s)=E(-\mathbf{k},s),$9

implies zero transverse charge current but finite transverse spin conductance,

$E(\mathbf{k},s)=E(-\mathbf{k},-s).$0

so that a pure transverse spin current flows parallel to the interface (Salehi et al., 2024). The same model exhibits an indirect conductance gap for $E(\mathbf{k},s)=E(-\mathbf{k},-s).$1 when the normal and p-wave Fermi circles no longer overlap (Salehi et al., 2024).

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

$E(\mathbf{k},s)=E(-\mathbf{k},-s).$2

while in 3D the corresponding driven system becomes a Weyl semimetal with spin-resolved Weyl nodes and Fermi-arc surface states (Liu et al., 25 Aug 2025). The same work emphasizes that the direction of spin splitting is perpendicular to the dimerization direction and can be controlled by the drive geometry (Liu et al., 25 Aug 2025).

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 $E(\mathbf{k},s)=E(-\mathbf{k},-s).$3-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 (Gong et al., 22 May 2026). The important point is not a complete replacement of even-wave altermagnetism, but a band- and spin-component-resolved coexistence of $E(\mathbf{k},s)=E(-\mathbf{k},-s).$4-, $E(\mathbf{k},s)=E(-\mathbf{k},-s).$5-, and $E(\mathbf{k},s)=E(-\mathbf{k},-s).$6-wave characters (Gong et al., 22 May 2026).

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” (Khodas et al., 27 Jan 2026). This is tied to the relation $E(\mathbf{k},s)=E(-\mathbf{k},-s).$7 and to the large non-relativistic spin splitting (Khodas et al., 27 Jan 2026). Separately, in three-dimensional $E(\mathbf{k},s)=E(-\mathbf{k},-s).$8-wave altermagnetic metals, the altermagnetic spin splitting can stabilize chiral $E(\mathbf{k},s)=E(-\mathbf{k},-s).$9-wave superconducting states under strong altermagnetic fields and high electron densities (Cadez et al., 26 Feb 2026). 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 (Das et al., 2024).

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 (Das et al., 2024). CeNiAsO is highlighted as a predicted p-wave magnet in the spin-group review (Jungwirth et al., 2024). Mn$[C_2\parallel \mathbf{t}]$0Si$[C_2\parallel \mathbf{t}]$1 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 (Sivianes et al., 2024). 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 $[C_2\parallel \mathbf{t}]$2, with electrically switchable p-wave order (Priessnitz et al., 19 Mar 2026). Wurtzite MnTe enters as the ferroelectric relativistic case with band-selective p-wave magnetism (Gong et al., 22 May 2026).

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 (Das et al., 2024). Photogalvanic probes are symmetry-selective in weak-SOC p-wave magnets: in Mn$[C_2\parallel \mathbf{t}]$3Si$[C_2\parallel \mathbf{t}]$4, 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 (Sivianes et al., 2024). 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 (Sukhachov et al., 2024). 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 (Salehi et al., 2024, Mizoguchi et al., 10 Mar 2026, Priessnitz et al., 19 Mar 2026).

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 $[C_2\parallel \mathbf{t}]$5 cases (Das et al., 2024, Liu et al., 25 Aug 2025), whereas others treat p-wave magnets as odd-parity, non-collinear analogs beyond strict altermagnetism (Jungwirth et al., 2024), or as time-reversal-symmetric p-wave magnets and antialtermagnets (Priessnitz et al., 19 Mar 2026). A cautious synthesis is therefore that p-wave altermagnets are best understood as the $[C_2\parallel \mathbf{t}]$6, 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.