Momentum-Alternating Spin Splitting in Solids

Updated 20 December 2025

Momentum-alternating spin splitting is a phenomenon where the spin splitting changes sign systematically across the Brillouin zone due to crystal and magnetic symmetries.
It manifests in both bulk and engineered low-dimensional systems with d-wave, g-wave, and checkerboard patterns, often emerging without strong spin–orbit coupling.
Experimental techniques like ARPES and theoretical models such as tight-binding and ab initio simulations validate its role in advancing novel spintronic applications.

Momentum-alternating spin splitting refers to a class of symmetry-allowed, often large-amplitude, spin splittings in crystalline solids where the sign and magnitude of the spin splitting vary systematically throughout the Brillouin zone (“alternate”), dictated by crystal, magnetic, and – in some cases – orbital structure. In contrast to trivial Zeeman, Rashba, or Dresselhaus effects, momentum-alternating spin splitting often emerges in materials with compensated magnetic order (i.e., net zero magnetization), where electronic bands remain spin-split at most k-points. This mechanism underlies the recently identified class of altermagnetic materials, yielding non-relativistic, robust, and symmetry-protected spin textures that persist even in the absence of strong spin–orbit coupling. The subject encompasses theoretical classification, tight-binding and ab initio modeling, multipolar formalism, experimental ARPES/SARPES signatures, and a growing body of proposals for spintronic applications.

1. Phenomenology and Symmetry Foundations

Momentum-alternating spin splitting (MASS) arises when the electronic band structure of a solid supports spin splitting that is an odd or sign-changing function of momentum, as dictated by the underlying magnetic space group, crystal symmetry, and the real-space arrangement of magnetic moments. Formally, the band energy difference between spin projections ΔE(k) = E↑(k) – E↓(k) satisfies ΔE(–k) = –ΔE(k) or, in some cases, alternates sign upon symmetry operations like π/2 rotations.

Canonical Zeeman splitting yields k-independent ΔE, while Rashba or Dresselhaus splitting yields ΔE ∝ ±|k|, with fixed chirality. In altermagnets and related systems, however, symmetry operations (such as time-reversal combined with spatial rotation or inversion) pair E↑(k) with E↓(A k) for some nontrivial point group operation A, enforcing sign-alternating patterns of spin splitting across momentum space (Šmejkal et al., 2021, Lin, 12 Mar 2025).

Minimal k·p Hamiltonians for such classes take the form:

$H(\mathbf{k}) = \varepsilon_0(\mathbf{k}) I_{2\times2} + d_z(\mathbf{k})\,\sigma_z$

with $d_z(\mathbf{k})$ an even or odd function depending on the symmetry, typically quadratic or higher in momentum:

$d_z(\mathbf{k}) \sim J\,k_x k_y \quad \text{(d-wave)},\quad d_z(\mathbf{k}) \sim J'\,k_x k_y (k_x^2 - k_y^2) \quad \text{(g-wave)}$

Spin splitting is then directly controlled by the parity and crystal-harmonic content of $d_z(\mathbf{k})$ , producing alternating sign structures in the Brillouin zone (Šmejkal et al., 2021, Guo et al., 2022).

2. Mechanisms: Exchange, Multipolar, and Nonrelativistic Origins

Nonrelativistic Antiferromagnetic Origin

The archetype for MASS is in compensated collinear or noncollinear antiferromagnets with broken ΘI (time-reversal × inversion) symmetry and magnetic space group (MSG) type I or III. In such systems, local exchange fields vary periodically (AFM “superexchange”), yielding a real-space exchange field h(r) with Fourier components hybridizing spin and momentum. The associated Bloch Hamiltonian has spin splitting $\Delta(\mathbf{k})$ that is odd in k (Yuan et al., 2020, Yuan et al., 2021):

$H_{\mathrm{eff}}(\mathbf{k};\delta) = \varepsilon_0 + \frac{\hbar^2|\mathbf{k}|^2}{2m^*} I_2 + \mathbf{h}(\mathbf{k},\delta) \cdot \vec\sigma, \quad \mathbf{h}(\mathbf{k},\delta) = J\delta\,\mathbf{k}$

DFT studies show that even small symmetry-breaking displacements of a non-magnetic ligand sublattice (e.g., oxygens in NiO) trigger pronounced momentum-dependent splittings exceeding hundreds of meV (Yuan et al., 2021).

Magnetoelectric Multipole Formalism

A unified relativistic extension classifies MASS via local electric multipole differences between symmetry-inequivalent sites. The leading spin splitting reads (Acosta, 28 May 2025):

$\Delta_s(\mathbf{k}) = -2\mu_B\eta_0[\lambda_0\,\Delta Q_0 + \lambda_1(\Delta\vec{d} \cdot \mathbf{k}) + \lambda_2\,\Delta Q_{ij}k_i k_j + \ldots]$

l=0 (monopole): k-independent splitting at Γ
l=1 (dipole): linear-in-k, "spin-Zeeman"
l=2 (quadrupole): quadratic, as in altermagnets, leading to alternating sign ΔE(k) patterns

This connects odd-parity spin splitting, spin-Zeeman, and standard Zeeman and Rashba/Dresselhaus effects as limiting cases of a general spin-multipole interaction.

Orbital–Crystal-Field and Sublattice Current Mechanisms

Microscopically, the anisotropic local crystal field (often from staggered ligand environments) mediates the momentum-dependence of the splitting. Tight-binding models for d-wave altermagnets show that sublattice-staggered crystal field combined with staggered AFM exchange yields a momentum-dependent, sign-alternating spin splitting (Vila et al., 30 Oct 2024):

$\Delta_s(\mathbf{k}) \propto m \left[ \cos(k_x a) - \cos(k_y a) \right]$

Odd-parity altermagnetism can moreover arise from loop (sublattice) currents in bipartite lattices, as in the generalized Haldane–Hubbard model. The k-dependence of the spin splitting is controlled by the current-induced form factor $j(\mathbf{k})$ and admits nonrelativistic, odd-in-k MASS even in the absence of net magnetic moment (Lin, 12 Mar 2025).

3. Hamiltonian Classification and Realizations

Momentum-alternating spin splitting can present as “d-wave,” “g-wave,” or higher harmonic textures, depending on the local point group and symmetry operations relating magnetic sublattices. The six primary altermagnetic classes are (Šmejkal et al., 2021):

Texture	Symmetry	k-dependence	Example
P–2	Planar, 2/m	$k_x k_y$	RuO₂, La₂CuO₄
P–4	Planar, 4/mmm	$k_x k_y (k_x^2{-}k_y^2)$
P–6	Planar, 6/mmm	complex $k_x, k_y$ cubic
B–2	Bulk, 2/m	$k_x k_z$	CuF₂
B–4	Bulk, 4/mmm	$k_x k_z (k_x^2{-}3k_y^2)$
B–6	Bulk, 6/mmm	$(k_x^2{-}k_y^2) (k_y^2{-}k_z^2) (k_z^2{-}k_x^2)$

The symmetry mandates locations of nodal lines, sign changes, and overall winding number. Exemplary materials include compensated collinear c-type AFM rutile RuO₂ (P–2, with DFT-observed ΔE up to ∼1 eV), NiAs-type CrSb (B–4, with chiral magnon splitting in both LSWT and TDDFPT), and LaMnO₃ (P–2, moderate splitting) (Zhang et al., 17 Mar 2025, Guo et al., 2022, Šmejkal et al., 2021).

Noncoplanar antiferromagnets such as MnTe₂ exhibit a “plaid-like” texture, described by quadratic (second-order) k·p expansions with in-plane spin splitting terms $\propto \alpha k_z (k_y \sigma_x + k_x \sigma_y)$ . This leads to a checkerboard pattern of spin sign in the k_x–k_y plane at fixed k_z, with experimental confirmation via SARPES showing sign flips as predicted (Zhu et al., 2023).

4. Edge, 1D, Chiral, and Strain-tuned Manifestations

Beyond bulk compounds, MASS arises in engineered or low-D systems:

Strained Graphene: In zigzag nanoribbons, lattice deformations induce a valley-dependent pseudomagnetic field. The competition between QSH edge states and pseudomagnetic boundary modes leads to a k-dependent out-of-plane spin splitting, alternating at valley crossings and tunable via strain and width (Yang et al., 2013):

$\Delta E(k) = 2\lambda_{\text{SO}} + 2\tau(k) e v_F B_s W$

Chiral 1D Systems: InSeI, a helical 1D material, exhibits a purely linear-in-k spin splitting dictated by the handedness of the chain, its sign controlled by enantiomorph and tunable by strain:

$\Delta E(k) = 2\alpha k,\quad \langle S_z(k) \rangle \propto \mathrm{sgn}(k)$

Under 4% strain, ΔE at the conduction-band minimum can reach 0.11 eV (Zhao et al., 2023).

Plaid/Checkerboard Textures: Noncoplanar AFMs exhibit MASS that is quadratic (or higher) in momentum, with alternating sign dictated by mirror and rotational symmetry of the magnetization texture (Zhu et al., 2023).

5. Experimental Detection and Spectroscopic Signatures

The unambiguous identification of MASS requires momentum-resolved, often spin-resolved, probes:

Spin-resolved ARPES/SARPES: Direct measurement of spin-polarized, sign-alternating band splitting; verified in MnTe₂ for plaid-like splitting and in RuO₂, CrSb for d-/g-wave textures (Zhu et al., 2023, Guo et al., 2022).
Magnetic Linear Dichroism (MLD) and Kerr/Faraday effects: Optical selection rules and dichroic responses tied to the symmetry of the underlying in-plane or bulk spin texture (Vila et al., 30 Oct 2024).
Transport: Momentum-alternating splitting yields nonreciprocal, valley-dependent, and direction-dependent spin filtering and Hall responses. In d-wave altermagnets, gate-tunable real-space spin precession (spin transistor effect) is directly controlled by the MASS amplitude (Liu et al., 7 Nov 2025).
Inelastic Neutron Scattering: Chiral magnon splitting in altermagnetic insulators, notably CrSb, provides a bosonic analog of the electronic MASS, with both LSWT-predicted and TDDFPT-augmented (broadened) signatures (Zhang et al., 17 Mar 2025).

6. Material Engineering and Theoretical Generalizations

MASS may be realized and controlled in a broad range of material platforms:

Inverse design: Symmetry-guided search in MSG databases yields hundreds of candidate low-Z, nonrelativistic altermagnets with optimal band splitting (Yuan et al., 2020).
Multipole engineering: By targeting specific local electric multipoles and symmetry connectivities, one may prescribe the momentum dependence of the spin splitting (Acosta, 28 May 2025).
Orbital–spin locking and strain control: In orbital-degenerate d-wave systems, applied mechanical or electrostatic fields, sublattice potential design, and strain can tune the magnitude and pattern of MASS (Vila et al., 30 Oct 2024, Yang et al., 2013).
Topological Phases: In the Haldane–Hubbard model, MASS leads to an "ALM Chern insulator" with quantized Hall response in the absence of net magnetization, stabilized by the odd-parity splitting $\Delta(\mathbf{k})$ (Lin, 12 Mar 2025).

Key distinctions between momentum-alternating (altermagnetic) spin splitting and conventional Rashba/Dresselhaus mechanisms include:

Inversion of symmetry role: MASS does not require broken global inversion (I), but instead, broken combined symmetries (ΘI, ΘIT) and specific MSG types (Yuan et al., 2020).
SOC independence: In canonical altermagnets, MASS arises without requiring relativistic spin–orbit coupling; Zeeman-type or Rashba/Dresselhaus effects are subleading or symmetry-forbidden in certain MSG backgrounds (Yuan et al., 2021).
Functional form: While Rashba/Dresselhaus produce non-alternating, chiral, linear-in-k splitting (e.g., ΔE ∝ |k|), MASS yields sign-alternating splitting dictated by k² or higher polynomials (e.g., ΔE ∝ k_x k_y, etc.) (Šmejkal et al., 2021, Guo et al., 2022).
Robustness and control: The sign, amplitude, and nodal structure of MASS are highly robust to disorder, strain, and—via multipole engineering—can be tailored for bespoke spintronic functionality (Acosta, 28 May 2025, Liu et al., 7 Nov 2025).

References

(Šmejkal et al., 2021) Šmejkal et al., Altermagnetism: spin-momentum locked phase protected by non-relativistic symmetries
(Yuan et al., 2020) Yuan et al., Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling
(Guo et al., 2022) Guo et al., Spin-split collinear antiferromagnets: a large-scale ab-initio study
(Yuan et al., 2021) Guo et al., Strong influence of non-magnetic ligands on the momentum dependent spin splitting in antiferromagnets
(Acosta, 28 May 2025) Guo, Unity Magnetoelectric Mechanism for Spin Splitting in Magnets
(Vila et al., 30 Oct 2024) Park et al., Orbital-spin Locking and its Optical Signatures in Altermagnets
(Liu et al., 7 Nov 2025) Liu et al., Altermagnetic Spin Precession and Spin Transistor
(Zhu et al., 2023) Yi et al., Observation of plaid-like spin splitting in a noncoplanar antiferromagnet
(Zhang et al., 17 Mar 2025) Lin et al., Chiral magnon splitting in altermagnetic CrSb from first principles
(Lin, 12 Mar 2025) Xie et al., Odd-parity altermagnetism through sublattice currents: From Haldane-Hubbard model to general bipartite lattices
(Zhao et al., 2023) Guo et al., Chirality-induced spin splitting in 1D InSeI
(Yang et al., 2013) Pan et al., Spin Splitting Induced by a Competition between Quantum Spin Hall Edge States and Valley Edge States

In summary, momentum-alternating spin splitting is the cornerstone of altermagnetism and related phenomena, offering symmetry-rich, tunable, and robust nonrelativistic spin structure in a wide materials landscape. MASS enables direct momentum-dependent spin selectivity in bulk solids, engineered heterostructures, and low-dimensional systems, and is verified both by theoretical modeling and direct spectroscopic observation. Its multipolar underpinning and symmetry-classification unify and generalize all known spin-splitting effects in solids, with significant implications for spintronic device design and the exploration of new quantum phases.