SOC-NCMs: Spin-Orbit Coupling in Noncentrosymmetric Metals

• SOC-NCMs are metals where relativistic spin–orbit coupling combined with broken inversion symmetry creates distinct spin-split or angular-momentum–split Fermi surfaces.
• Symmetry-based analyses and DFT methods reveal various splitting mechanisms (Rashba, Dresselhaus, Weyl, Ising) that govern the complex band topology in materials like NbP.
• The phenomena drive novel effects such as anomaly-induced transport, low-energy plasmons, nonlinear spin responses, and parity-mixed superconductivity, impacting device applications.

Spin–orbit-coupled noncentrosymmetric metals (SOC-NCMs) are metals or semimetals in which relativistic spin–orbit coupling acts in a crystal environment without an inversion center. In such systems, the combined action of SOC and inversion-symmetry breaking lifts spin degeneracies over extended regions of momentum space, reorganizes the band structure into spin-split or total-angular-momentum-split Fermi surfaces, and produces low-energy responses that are strongly shaped by Berry curvature, orbital magnetic moment, gyrotropy, and multiband compensation. The resulting phenomenology spans conventional Rashba- and Dresselhaus-type splittings, higher-multipole spin-orbit textures, Kramers nodal lines and Kramers Weyl points, low-energy plasmons, Edelstein magnetoelectricity, anomaly-driven nonlinear transport, and parity-mixed superconductivity (Ahn et al., 2015, Yang et al., 12 Feb 2026, Bahari et al., 20 Dec 2025).

1. Symmetry principles and the definition of SOC-NCMs

At low energy, the generic single-band description of a nonmagnetic noncentrosymmetric metal with time-reversal symmetry is

$$H(\mathbf{k})=\varepsilon_0(\mathbf{k})+\mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma},\qquad E_{\pm}(\mathbf{k})=\varepsilon_0(\mathbf{k})\pm|\mathbf{g}(\mathbf{k})|,$$

with $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$. Broken inversion symmetry is therefore the necessary condition for nonzero spin splitting, while time-reversal symmetry still enforces Kramers degeneracy at time-reversal-invariant momenta (Yang et al., 12 Feb 2026).

A symmetry-based classification to linear order in $\mathbf{k}$ identifies four universal SOC spin-splitting types in nonmagnetic inversion-broken crystals: Rashba, Dresselhaus, Weyl, and Ising. Rashba SOC takes the familiar form $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$ in polar crystals. Weyl-type SOC corresponds to $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$, yielding a hedgehog texture. Ising SOC locks spin to a crystal axis, while Dresselhaus symmetry admits both linear and cubic forms, including the zinc-blende expression

$$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$

These are not merely alternative parametrizations: they correspond to distinct inversion-odd irreducible representations of the parent point groups and therefore imply distinct nodal structures, spin textures, and transport tensors (Yang et al., 12 Feb 2026).

In multiorbital and heavy-element systems, an effective spin-$1/2$ description can be insufficient. A symmetry-based $j$-multiplet theory for $j\in\{1/2,3/2,5/2\}$ shows that for $j>1/2$, the SOC Hamiltonian contains higher-rank tensor operators in addition to the usual modified Rashba term. In that regime, the relevant internal degree of freedom is total angular momentum rather than pure spin, and the Bloch states carry band-dependent total-angular-momentum textures classified by vorticities $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$0, rather than a single helicity winding (Bahari et al., 20 Dec 2025).

A further symmetry-based topological narrative distinguishes chiral and achiral noncentrosymmetric metals. In this construction, chiral crystals with SOC realize Kramers Weyl semimetals, whereas achiral noncentrosymmetric crystals with SOC realize Kramers nodal line metals. The latter host doubly degenerate Kramers nodal lines connecting time-reversal-invariant momenta, enforced by time reversal together with mirror or roto-inversion symmetries (Xie et al., 2020). This suggests that the topological content of SOC-NCMs is often fixed at the level of crystal symmetry rather than by accidental band inversion alone.

2. Band structures, Fermi surfaces, and representative electronic structures

The prototypical SOC-NCM case study is NbP, a body-centered tetragonal, noncentrosymmetric semimetal in space group $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$1 (No. 109), with $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$2 Å and $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$3 Å. In generalized-gradient DFT, NbP is a compensated semimetal with valence–conduction overlaps of about $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$4 meV, a pseudogap centered $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$5 meV below $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$6, and an extremely small density of states $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$7 states/eV per formula unit (Ahn et al., 2015).

Without SOC, mirror symmetry protects nodal loops in the $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$8 and $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$9 mirror planes. Including SOC in a noncentrosymmetric lattice produces two simultaneous effects: local anticrossings of a few tens of meV near $\mathbf{k}$0, and a global lifting of spin degeneracy over extended symmetry planes, with splittings from about $\mathbf{k}$1 meV up to $\mathbf{k}$2 meV depending on $\mathbf{k}$3. In NbP, these SOC-induced splittings destroy the nodal loops and generate $\mathbf{k}$4 Weyl points near, but not on, the Fermi surface. Because the $\mathbf{k}$5 SOC of Nb is modest, the resulting Fermi surfaces still map out the original nodal-loop geometry, an intermediate regime between line-node semimetal and fully SOC-reorganized Weyl semimetal (Ahn et al., 2015).

The Fermiology of NbP is unusually delicate. Stoichiometric NbP has equal electron and hole densities of $\mathbf{k}$6 carriers per formula unit, with Fermi velocities $\mathbf{k}$7 cm/s and $\mathbf{k}$8 cm/s, and directional plasma energies $\mathbf{k}$9 eV and $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$0 eV. The spin-split pockets form closely spaced “Russian doll nested” surfaces with a pinched-torus topology that decomposes into “boomerang” hole pockets and “new moon” electron pockets. This morphology is a direct consequence of inversion breaking plus SOC, not merely of Weyl-node placement (Ahn et al., 2015).

More generally, the local little group at a given $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$1-point determines the form of the SOC splitting. In noncentrosymmetric half-Heusler compounds such as CoZrBi and SiLiIn, symmetry-adapted $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$2 analysis finds linear Dresselhaus splitting at the $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$3 point with $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$4 symmetry, Rashba splitting with linear and cubic terms at the $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$5 point with $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$6 symmetry, a SOC-enabled Zeeman-like splitting at the non-time-reversal-invariant $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$7 point, and band splitting with vanishing spin polarization along $$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$8–$$\mathbf{g}_R(\mathbf{k})\propto \mathbf{k}\times \hat{\mathbf{n}}$$9 because the little group there is non-pseudo-polar (Dutta et al., 2023). This local-symmetry perspective clarifies that SOC-NCM band structures are often patchworks of distinct spin-splitting mechanisms rather than realizations of a single global Rashba model.

The same point becomes sharper in locally noncentrosymmetric crystals. In multi-sublattice systems, the sublattice degree of freedom permits an even-parity symmetric quasi-spin-orbit coupling on inter-sublattice bonds and an odd-parity antisymmetric quasi-spin-orbit coupling on intra-sublattice bonds. Neither term alone necessarily produces a global spin splitting, but their coexistence does. This sublattice-resolved mechanism generalizes the usual single-sublattice ASOC paradigm and is directly visible in quasiparticle-interference signatures (Biderang et al., 2019).

3. Topological structure, polarization, and collective modes

A central topological result is that in achiral noncentrosymmetric metals with SOC, the most general two-band Hamiltonian near an achiral time-reversal-invariant momentum can enforce a zero eigenvalue of the linear SOC matrix, producing a Kramers nodal line. These nodal lines connect time-reversal-invariant momenta and generate two characteristic Fermi-surface topologies: the spindle torus and the octdong. In the octdong case, fixed-$H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$0 slices are described by two-dimensional massless Dirac Hamiltonians, which in thin films yield quantized optical conductance plateaus and in bulk give linear optical conductivity with zero-frequency onset (Xie et al., 2020).

For continuum SOC-NCM models with isotropic $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$1 splitting, the dynamical polarization function can be evaluated exactly and within long-wavelength expansion. The RPA dielectric function,

$H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$2

shows that the intra- and interband particle-hole continua are separated by a finite gap at $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$3. Above the band-touching point, the interband continuum width is $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$4; below it, the width becomes $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$5. This difference is a direct signature of the Fermi-surface topology change across the band-touching point. In both regimes there is a single undamped optical plasmon between the continua, with a long-wavelength dispersion proportional to $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$6, but the plasmon below the band-touching point has a smaller velocity than the one above it (Verma et al., 2020).

NbP realizes a more material-specific variant of this electrodynamics. Its SOC-split nested Fermi surfaces produce low-energy collective electron–hole excitations in the $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$7–$H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$8 meV range. Within a loss-function calculation using $H_W=\alpha_W\,\boldsymbol{\sigma}\cdot\mathbf{k}$9 meV, the predicted modes are a $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$0-axis plasmon near $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$1 meV and in-plane plasmons near $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$2 meV and $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$3 meV. These modes arise because SOC transfers spectral weight from the Drude channel into low-energy interband transitions between the nested Fermi sheets (Ahn et al., 2015).

A common misconception is that gyrotropic optical activity in a noncentrosymmetric metal necessarily implies Kerr rotation in reflection. The corrected electrodynamics of gyrotropic metals shows otherwise. The conductivity tensor contains a linear-in-wavevector term,

$$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$4

and the circular refractive indices satisfy

$$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$5

This produces natural optical activity in transmission. However, once the full gyrotropy current,

$$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$6

is used in the boundary conditions, the reflection coefficients for right- and left-circular polarization are equal, $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$7, so the polar Kerr angle vanishes in a time-reversal-symmetric gyrotropic metal at normal incidence (Mineev et al., 2010). The distinction between transmission gyrotropy and reflection Kerr response is therefore essential in SOC-NCM optics.

4. Magnetoelectricity, anomaly-driven transport, and nonlinear responses

In multiorbital noncentrosymmetric systems, current-induced magnetization is controlled not only by spin but also by orbital texture. A superconducting Edelstein analysis with orbital Rashba coupling and atomic SOC finds that the orbital Edelstein effect is generically larger than the spin Edelstein effect, with $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$8 typically about an order of magnitude smaller than $$H_D=\gamma\big[k_x(k_y^2-k_z^2)\sigma_x+k_y(k_z^2-k_x^2)\sigma_y+k_z(k_x^2-k_y^2)\sigma_z\big].$$9. The sign and magnitude are governed primarily by the number of Fermi-surface bands and their mirror parity, not by the sign of the atomic SOC constant $1/2$0. In particular, the orbital response changes sign robustly across avoided crossings between bands of opposite mirror parity, while the spin response is strongly enhanced when multiple same-parity bands cross the Fermi level (Ando et al., 2024). Although formulated for superconductors, the same momentum-space selection rules carry over to normal-state SOC-NCMs in the $1/2$1 limit.

SOC-NCMs also support a second-order spin current under an ordinary electric field without requiring magnetic order. In Boltzmann theory, a uniform electric field generates a nonlinear spin current proportional to $1/2$2 and $1/2$3, while time-reversal symmetry forbids the corresponding even-order charge current. In Rashba and Dresselhaus settings this response is controlled by the Fermi-surface spin texture and can be rectified into a dc spin current under ac or terahertz driving (Hamamoto et al., 2017). This establishes a broad nonlinear spin-transport channel intrinsic to inversion-broken SOC metals.

A major theoretical reorientation concerns the chiral anomaly. In SOC-NCMs, several transport analyses argue that anomaly physics is not exclusively a Weyl-node property but can instead be a Fermi-surface property. In minimal SOC-NCM models, the two spin-split Fermi surfaces around a single touching point carry opposite Berry-curvature flux but the same-sign orbital magnetic moment. This yields a longitudinal magnetoconductance whose sign remains positive in the semiclassical regime, unlike in Weyl metals where the sign can reverse with increasing internode scattering. The same-sign orbital magnetic moment also changes the planar Hall and geometrical Hall responses, making them qualitatively distinct from Weyl systems (K et al., 2023).

The nonlinear anomalous Hall sector strengthens this distinction. For three-dimensional chiral quasiparticles, SOC-NCMs exhibit a chiral-anomaly-induced nonlinear anomalous Hall conductivity that is driven by the orbital magnetic moment and remains consistently negative regardless of interband scattering strength. It scales quadratically with magnetic field, in contrast to the linear-$1/2$4 dependence in Weyl semimetals, and Zeeman coupling acts as an effective tilt that can further enhance the response (Ahmad et al., 2024). A closely related semiclassical treatment of the chiral-anomaly-induced nonlinear Hall effect in SOC-NCMs reaches the same qualitative conclusion: the untilted response is quadratic in $1/2$5 and negative for both nonmagnetic and magnetic impurities, while band tilt introduces a direction-dependent linear-in-$1/2$6 correction and weak or strong sign-reversal behavior controlled by the angle between the field and the tilt vector (K et al., 1 Aug 2025).

Thermomagnetic anomaly responses follow similar logic. Strain in a SOC-NCM induces SOC anisotropy and an axial electric field $1/2$7, even when the corresponding axial magnetic field vanishes. In the presence of a magnetic field, the anomaly term $1/2$8 generates Ettingshausen temperature gradients whose angular structure separates conventional Lorentz-force channels from Berry-curvature-driven anomalous channels. In the open-circuit geometry, $1/2$9, $j$0, and $j$1, with $j$2 capable of sign reversal as interband scattering changes, while the longitudinal anomaly channel remains quadratic in $j$3 and fixed in sign (K et al., 18 Aug 2025).

These results collectively imply that SOC-NCM transport is best understood as a hierarchy of Fermi-surface phenomena: Berry curvature, orbital magnetic moment, and interband scattering redistribute weight between spin-split sheets and generate responses that can resemble Weyl transport in form but differ in sign structure, magnetic-field scaling, and symmetry control.

5. Superconducting SOC-NCMs and parity-mixed pairing

In a noncentrosymmetric superconductor, antisymmetric SOC permits the mixed-parity gap structure

$j$4

so parity is no longer a good quantum number and singlet and triplet components generally admix. In the strong-ASOC limit, the protected triplet channel satisfies $j$5, and the helicity-band gaps are $j$6. This structure underlies the magnetoelectric effect, helical superconducting phases, finite-momentum pairing, and both fully gapped and nodal topological superconductivity in noncentrosymmetric systems (Smidman et al., 2016).

The superconducting state inherits the same inversion-breaking electrodynamics discussed in the normal state, but with coherence-factor modifications. The corrected gyrotropic boundary problem remains reciprocal so long as time-reversal symmetry is preserved, which means natural optical activity can survive in transmission while Kerr rotation in reflection remains absent unless time-reversal symmetry is broken (Mineev et al., 2010). This point is relevant because superconducting gyrotropy is often conflated with Kerr activity.

A concrete strong-SOC superconducting platform is provided by NbIr$j$7B$j$8 and TaIr$j$9B$j\in\{1/2,3/2,5/2\}$0, both monoclinic noncentrosymmetric superconductors in space group $j\in\{1/2,3/2,5/2\}$1 (No. 9). In DFT with SOC, their bands near $j\in\{1/2,3/2,5/2\}$2 are predominantly Ir $j\in\{1/2,3/2,5/2\}$3-derived and show ASOC-induced splittings from about $j\in\{1/2,3/2,5/2\}$4 meV up to $j\in\{1/2,3/2,5/2\}$5 meV. Experimentally, $j\in\{1/2,3/2,5/2\}$6 is about $j\in\{1/2,3/2,5/2\}$7 K in NbIr$j\in\{1/2,3/2,5/2\}$8B$j\in\{1/2,3/2,5/2\}$9 and $j>1/2$0 K in TaIr$j>1/2$1B$j>1/2$2, and the zero-temperature upper critical fields exceed the Pauli limit in both compounds. NbIr$j>1/2$3B$j>1/2$4 additionally shows a heat-capacity jump $j>1/2$5 and is better fit by a two-gap $j>1/2$6 model than by a single-gap one, while TaIr$j>1/2$7B$j>1/2$8 lies near the weak-coupling value $j>1/2$9. These data indicate strong ASOC and parity-mixed superconductivity in a low-symmetry setting (Górnicka et al., 2021).

Locally noncentrosymmetric superconductors extend this logic. In a two-sublattice setting, inter-sublattice symmetric quasi-spin-orbit coupling supports odd-parity triplet pairing, while intra-sublattice antisymmetric quasi-spin-orbit coupling supports parity-mixed singlet–triplet pairing. Quasiparticle interference then resolves whether superconductivity is dominated by inter-sublattice triplet or intra-sublattice $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$00 structure, and whether the relevant normal-state quasi-spin-orbit coupling is symmetric, antisymmetric, or both (Biderang et al., 2019). This suggests that the notion of “noncentrosymmetric superconductivity” is not exhausted by globally inversion-broken crystals; locally inversion-broken multicomponent systems can realize closely related physics through a different microscopic route.

6. Spin relaxation, correlated limits, and materials scope

Spin relaxation in SOC-NCMs is not confined to the textbook Dyakonov–Perel limit. A unified two-band treatment with both intra-band and inter-band SOC gives

$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$01

where $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$02, $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$03 is the inversion-breaking intra-band SOC, $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$04 the inter-band spin-admixture matrix element, and $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$05 and $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$06 the Zeeman and band-separation scales. This formula recovers both Dyakonov–Perel and Elliott–Yafet limits and shows explicit crossovers between them, including in inversion-broken states that become Elliott–Yafet-like when $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$07 (Boross et al., 2012). The result is significant because it places SOC-NCM spin dynamics on the same footing as band splitting, rather than treating relaxation as a separate phenomenology.

In the strongly correlated limit, the same inversion-breaking SOC that splits metallic bands generates anisotropic spin exchange in Mott states. A half-filled SOC Hubbard model with spin-dependent hopping

$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$08

produces, in strong coupling, Heisenberg exchange, Dzyaloshinskii–Moriya exchange $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$09, KSAEW anisotropy, ring exchange, and several four-spin chiral terms. The second-order couplings scale as $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$10, while fourth-order terms scale as $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$11, becoming important as $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$12 decreases toward the Mott transition. This establishes a continuous bridge between metallic SOC-NCM behavior and chiral magnetism in correlated insulators (Makuta et al., 7 Apr 2025).

The materials landscape is correspondingly broad. Prototypical platforms include nodal-loop and Weyl-semimetal members such as NbP (Ahn et al., 2015); polar and giant-Rashba compounds such as BiTeI and GeTe; chiral fermion materials discussed in Kramers-Weyl and multipolar-SOC settings (Xie et al., 2020, Bahari et al., 20 Dec 2025); oxide and polar-metal heterostructures with orbital Rashba coupling (Ando et al., 2024); half-Heuslers with point-dependent Dresselhaus, Rashba, and Zeeman-like splitting (Dutta et al., 2023); and low-symmetry intermetallic superconductors such as NbIr$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$13B$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$14 and TaIr$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$15B$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$16 (Górnicka et al., 2021).

Several recurring interpretive points follow from this literature. First, simple Rashba or Dresselhaus language is often insufficient once the low-energy manifold is multiorbital, high-$\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$17, or locally noncentrosymmetric. Second, topological response in SOC-NCMs is frequently encoded in Fermi-surface geometry and Berry flux rather than in isolated Weyl nodes alone. Third, optical, thermoelectric, and nonlinear transport signatures are exceptionally sensitive to compensation, band crossings, and interband scattering. This suggests that quantitative modeling of SOC-NCMs requires dense $\mathbf{g}(-\mathbf{k})=-\mathbf{g}(\mathbf{k})$18-space sampling, explicit symmetry analysis, and careful treatment of interband processes, particularly in materials such as NbP where the fine structure of the spin-split Fermi surface is itself the primary observable (Ahn et al., 2015).