Néel-Order-Locked Polarization

• Néel-order-locked polarization is a phenomenon where the polarization (electric, optical, spin, or valley) is directly controlled by the antiferromagnetic Néel vector via symmetry-allowed couplings.
• It unifies various responses across rare-earth orthochromites, altermagnets, and bilayer systems, linking magnetic order to measurable properties like Kerr rotation and spin accumulation.
• Different material systems employ mechanisms such as exchange striction or symmetry breaking, with techniques like DFT, Berry-phase, and magneto-optical microscopy used to quantify the locked polarization.

Néel-order-locked polarization denotes a class of phenomena in which a polarization channel is constrained by the Néel vector of an antiferromagnet or altermagnet, so that reorienting the magnetic order reorients, switches, or suppresses the polarization. In recent literature, the locked quantity is not limited to ferroelectric polarization: it includes spontaneous electric polarization in type-II multiferroics, spin-layer and valley polarization in symmetry-locked bilayer altermagnets, Kerr rotation and ellipticity arising from antisymmetric optical conductivity, and interfacial or supercurrent-induced spin polarization. Across these settings, the defining feature is a symmetry-allowed coupling between a polar observable and the antiferromagnetic order parameter, linear in some realizations and quadratic in others (Guo et al., 4 May 2025, Pan et al., 8 Dec 2025, Zhang et al., 28 Feb 2026, Jiang et al., 29 Aug 2025).

1. Conceptual scope and formal structure

The term is used across several neighboring subfields to describe different observables that are all tied to the orientation of antiferromagnetic order. In some systems the locked quantity is a true electric polarization $\mathbf P$; in others it is an optical, valley, layer, or spin polarization. What unifies these cases is not the microscopic origin but the one-to-one dependence on the Néel order parameter imposed by symmetry or by an exchange-mediated locking mechanism.

Realization	Locked observable	Representative relation
Rare-earth orthochromites (Rajeswaran et al., 2012)	field-induced switchable electric polarization	$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$
2D altermagnetic multiferroics (Guo et al., 4 May 2025)	spontaneous electric polarization	$P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$
Symmetry-locked bilayer altermagnet (Zhang et al., 28 Feb 2026)	spin-layer or valley polarization	$P_i=\alpha_i(\epsilon)\,n_i$
Hematite MOKE (Pan et al., 8 Dec 2025)	Kerr response via antisymmetric optical conductivity	$$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$
AFI/FM bilayer (Jiang et al., 29 Aug 2025)	out-of-plane spin polarization	$\sigma_z$ is largest when $\boldsymbol{\mu}_s\parallel\mathbf n$
Superconductor/altermagnet hybrid (Vakili et al., 23 Mar 2026)	supercurrent-induced spin polarization	$$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$

A recurrent source of confusion is terminological. The same label is applied to electric, optical, momentum-space, and spin polarizations. This suggests that the term functions as a unifying symmetry descriptor rather than as a restriction to ferroelectric order alone.

2. Precursor phenomenology in rare-earth orthochromites

An important precursor appears in weakly ferromagnetic rare-earth orthochromites $R$CrO$_3$, where field-induced switchable polarization emerges below the chromium Néel temperature only when the rare-earth ion is magnetic (Rajeswaran et al., 2012). In $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$0 Sm, Gd, Tb, Tm, and Er, a spontaneous polarization $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$1–$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$2 appears on cooling below $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$3 in the range $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$4–$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$5. Pyroelectric-current measurements under $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$6 poling of approximately $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$7–$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$8 show true sign reversal of $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$9, and PUND pulses confirm an intrinsic remanent polarization of approximately $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$0 under the small PUND effective field.

The temperature dependence is locked to chromium ordering. The polarization onset coincides with $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$1 and grows on further cooling roughly as

$P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$2

with experimental $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$3–$P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$4. In ErCrO$P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$5, which undergoes a spin-reorientation or Morin transition at $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$6, the polarization collapses to zero below $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$7 exactly when the weak ferromagnetism of the Cr sublattice disappears. Application of magnetic field $P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$8–$P^\alpha=\Pi^\alpha_{\beta\gamma}L^\beta L^\gamma$9 suppresses the spin reorientation, revives the weak-ferromagnetic state, and restores $P_i=\alpha_i(\epsilon)\,n_i$0 down to the lowest temperature. Isothermal scans give a roughly linear magnetoelectric coefficient $P_i=\alpha_i(\epsilon)\,n_i$1–$P_i=\alpha_i(\epsilon)\,n_i$2, with weak higher-order curvature.

The symmetry argument is local-to-global. The rare-earth ion sits on a mirror plane with point group $P_i=\alpha_i(\epsilon)\,n_i$3, so a local electric dipole is allowed; however, in ideal Pbnm the eight $P_i=\alpha_i(\epsilon)\,n_i$4 sites in the unit cell are related by inversion and mirror operations, yielding an antiferroelectric cancellation. The proposed mechanism is that the external poling field $P_i=\alpha_i(\epsilon)\,n_i$5 breaks macroscopic inversion symmetry and seeds a small polar distortion, while the exchange field from the canted Cr sublattice,

$P_i=\alpha_i(\epsilon)\,n_i$6

acts uniformly on the rare-earth ions and stabilizes one polarity. In the schematic Landau expansion,

$P_i=\alpha_i(\epsilon)\,n_i$7

minimization gives

$P_i=\alpha_i(\epsilon)\,n_i$8

Here the one-to-one correspondence between polarization and weak ferromagnetism anticipates later NOLP formulations, although the later altermagnetic terminology was not yet in use.

3. Electric polarization in altermagnetic type-II multiferroics

Guo et al. developed the most explicit symmetry-based formulation of Néel-order-locked electric polarization in altermagnetic type-II multiferroics (Guo et al., 4 May 2025). The starting point is the contrast between a conventional collinear antiferromagnet with combined $P_i=\alpha_i(\epsilon)\,n_i$9 symmetry and an altermagnet. In the conventional case, the two magnetic sublattices $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$0 and $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$1 are related by inversion, implying equal and opposite local dipoles and hence $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$2. In the altermagnetic case, the antiparallel spins on $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$3 and $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$4 are not connected by inversion. Writing the local dipole on sublattice $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$5 as

$$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$6

the total dipole per cell becomes

$$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$7

Because inversion is absent in $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$8, nothing enforces $$\boldsymbol{\sigma}^A=\alpha(\omega)\,(\hat{\mathbf c}\times\mathbf N)$$9, and generically $\sigma_z$0. The resulting locked polarization is therefore

$\sigma_z$1

For two-dimensional layer groups with two magnetic sites per cell, the analysis yields exactly eight distinct locking behaviors $\sigma_z$2. The magnetoelectric coupling is

$\sigma_z$3

Category	Layer groups / site symmetry	$\sigma_z$4
1	LG 19, 20, 21 / ..2	$\sigma_z$5
2	LG 50 / 2..	$\sigma_z$6
3	LG 53, 54, 76 / 4.. or 222.	$\sigma_z$7
4	LG 57, 58 / 2.mm etc.	$\sigma_z$8
5	LG 59, 60 / 2mm.	$\sigma_z$9
6	LG 67 / 3..	$\boldsymbol{\mu}_s\parallel\mathbf n$0
7	LG 68 / 3..	$\boldsymbol{\mu}_s\parallel\mathbf n$1
8	LG 79 / -6..	$\boldsymbol{\mu}_s\parallel\mathbf n$2

In every category, the functional form is fixed by symmetry once the layer group is specified; the constants $\boldsymbol{\mu}_s\parallel\mathbf n$3 are material dependent. The components of $\boldsymbol{\mu}_s\parallel\mathbf n$4 are either $\boldsymbol{\mu}_s\parallel\mathbf n$5- or $\boldsymbol{\mu}_s\parallel\mathbf n$6-periodic in $\boldsymbol{\mu}_s\parallel\mathbf n$7.

Monolayer MgFe$\boldsymbol{\mu}_s\parallel\mathbf n$8N$\boldsymbol{\mu}_s\parallel\mathbf n$9 is presented as a prototypical realization. It crystallizes in LG 59 with point group $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$0 and two Fe sublattices. DFT using VASP + PAW + PBE finds that the collinear altermagnetic Néel order is lower in energy than the ferromagnetic state by approximately $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$1. The electronic bands show momentum-dependent spin splitting up to approximately $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$2 at certain $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$3, both without and with SOC. Using the modern theory of polarization, the Berry-phase calculation yields an out-of-plane polarization

$$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$4

which vanishes at $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$5 and changes sign between $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$6. The energy barrier for rotating $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$7 and switching $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$8 is below $$\mathbf P_s=-\,t_m\gamma\,(q_y\,\hat x+q_x\,\hat y)$$9.

The same work proposes magneto-optical microscopy as a domain probe. At fixed $R$0, the antisymmetric optical conductivities obey $R$1 and $R$2, while the Faraday-angle maximum occurs at

$R$3

This gives a one-to-one map from the optical response to the Néel angle and therefore to the sign and direction of the locked polarization.

4. Linear locking in symmetry-locked bilayer altermagnets

A distinct realization appears in the symmetry-locked bilayer altermagnet proposed to combine momentum-space topology with coupled antiskyrmions (Zhang et al., 28 Feb 2026). Here the staggered antiferromagnetic order is described by the Néel vector

$R$4

and the polarization channels are defined directly in terms of topological or Berry-curvature observables. The principal examples are the spin-layer polarization

$R$5

the valley polarizations

$R$6

and charge or orbital polarization, which vanishes at charge neutrality although layer-resolved versions can survive.

The locking is enforced by residual symmetry. In the unstrained bilayer, the overall point group is $R$7. With the A-AFM configuration and stacking operator $R$8, global inversion and one twofold rotation are broken, while the orthogonal mirror class is preserved. Under SOC, the allowed polar vector depends on the Néel-vector direction. For $R$9, the invariance group is $_3$0 itself and only $_3$1 is allowed. For $_3$2, the remaining subgroup forces $_3$3 and permits only $_3$4. For $_3$5, the allowed component is only $_3$6. The general form is therefore

$_3$7

or equivalently

$_3$8

with only the diagonal component along the residual symmetry axis remaining nonzero.

The phenomenology is summarized by the strain-dependent coupling

$_3$9

In the unstrained or compressive regime $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$00, first-principles calculations give $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$01, so $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$02 opens a spin-layer-polarized quantum spin Hall gap. The reported bandgap is $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$03 with spin Chern numbers $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$04, yielding

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$05

Under tensile strain $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$06, $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$07 and $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$08 become nonzero while $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$09 flips sign and may vanish. For $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$10, one layer enters a Chern-insulator phase while the other remains Weyl semimetal, generating a finite $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$11; the $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$12 case is analogous for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$13.

Because $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$14, rotating $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$15 by $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$16 toggles the polarization channel. In the QSH regime with $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$17 and $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$18, the system exhibits spin-layer polarization and two helical edge modes. Rotating $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$19 into the film plane enforces $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$20 and allows an in-plane channel, producing a spin-layer-polarized QAH + Weyl state together with a valley Hall response. In the same structure, the interlayer co-directional and locked in-plane Dzyaloshinskii-Moriya interactions stabilize coupled antiskyrmion pairs with compensated topological charges. Since $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$21, the total gyrovector vanishes and the Hall-like deflection is fully canceled, while the layer-resolved polarizations remain nonzero.

5. Magneto-optical Néel-order-locked polarization in hematite

In hematite, NOLP appears not as a static electric dipole but as a Kerr rotation and ellipticity locked to the Néel vector in a collinear antiferromagnetic insulator with broken $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$22 symmetry (Pan et al., 8 Dec 2025). The key observable is the antisymmetric optical conductivity,

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$23

which determines the Kerr angle through the usual Fresnel formula,

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$24

The central result is that the Néel-order term arises at first order in SOC, whereas the magnetization contribution is at least third order. With $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$25, one obtains to leading order

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$26

so that

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$27

By contrast, the weak ferromagnetic moment satisfies $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$28 with Moriya parameter $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$29, giving

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$30

The paper therefore concludes that the magnetization term is two orders of magnitude smaller than the Néel term in hematite.

The symmetry analysis is performed for space group $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$31 and collinear AFM order $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$32. For $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$33, a glide plane combined with time reversal constrains the allowed coupling to $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$34; for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$35, the response is correspondingly rotated; for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$36, the threefold and twofold rotations enforce $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$37. Collecting the cases gives

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$38

at first order in $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$39.

The measured polar-Kerr signal is of order milliradian and is dominated by the Néel contribution rather than by weak ferromagnetism or the applied field. In the visible range $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$40–$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$41, the calculated coefficient is

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$42

which yields

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$43

at $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$44 and a Kerr rotation of approximately $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$45. The ferromagnetic contribution is of order $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$46, and the field-induced term evaluated at $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$47 is another order smaller. This case directly refutes the common assumption that a sizeable MOKE signal necessarily tracks net magnetization.

6. Switching protocols and related spin-polarization extensions

Several recent works extend the locking idea from passive readout to active control of the Néel state and to spin-polarization channels that are not strictly electric. In parity-time-symmetric A-type antiferromagnetic multilayers, the two orientations $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$48 are exactly degenerate until symmetry-breaking light is applied (Xue et al., 18 Mar 2025). Under circularly polarized illumination, time reversal is broken by the handedness and inversion by nonreciprocal attenuation through the layered stack. The free-energy contrast

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$49

is nonzero only above the band gap and changes sign under $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$50 reversal or by tuning $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$51. In the six-layer $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$52 model, $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$53 for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$54, while for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$55 one finds $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$56 favoring $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$57 and $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$58 favoring $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$59. For bilayer MnBi$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$60Te$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$61, $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$62 and intensities $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$63 yield $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$64 up to $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$65, about twice the intrinsic anisotropy of approximately $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$66. For bilayer CrI$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$67, $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$68 and the threshold is $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$69 for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$70. A plausible implication is that any polarization channel locked one-to-one to the Néel vector should be optically switchable by the same handedness- and frequency-selective protocol.

A more direct spintronic realization appears in AFI/FM bilayers, where the adjacent antiferromagnetic insulator engineers a self-generated out-of-plane spin polarization in a single ferromagnet (Jiang et al., 29 Aug 2025). A charge current $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$71 in the FM produces a self-generated spin accumulation

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$72

subject to the spin-diffusion equation and spin-mixing boundary conditions at the AFI/FM interface. Reflection is maximal when the incident spin is collinear with the Néel vector, and interfacial spin-orbit coupling or spin swapping rotates the reflected spin to generate $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$73. The formal result is that $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$74 is largest when $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$75 is maximal. Experimentally, large out-of-plane torques are observed for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$76, which is precisely the geometry in which $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$77: the normalized ST-FMR ratios are

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$78

whereas for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$79 they fall to approximately $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$80 and $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$81. Field-free perpendicular switching occurs only for $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$82, with critical current density

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$83

An analogous extension to dissipationless transport is predicted for superconductor/$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$84-wave altermagnet hybrids (Vakili et al., 23 Mar 2026). Starting from a finite-$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$85 BdG Hamiltonian with Rashba SOC and momentum-dependent altermagnetic spin splitting, the first-order free-energy correction is

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$86

which produces the effective spin polarization

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$87

The corresponding field-like Néel torque is

$$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$88

The symmetry analysis shows that only the component of $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$89 perpendicular to $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$90 contributes, so rotating the Néel vector switches the effect on or off. For realistic parameters the estimated torque is approximately $$P \approx -(\alpha\,\mathbf L\times\mathbf M+\beta\,\mathbf H_{\rm ex})/a_P$$91, sufficient in the calculation to drive domain-wall motion or reverse the Néel orientation within a domain wall.

Taken together, the literature distinguishes at least two mechanistic routes. In rare-earth orthochromites, polarization is stabilized by poling-field symmetry breaking plus exchange striction from a weakly ferromagnetic canted sublattice. In altermagnets and related topological or optical systems, the locked response instead follows from the symmetry of the Néel order itself and can persist with vanishing net magnetization. NOLP is therefore neither synonymous with weak ferromagnetism nor guaranteed by antiferromagnetism alone: it appears only when the relevant inversion, mirror, or rotation constraints permit a polar channel to couple to the Néel vector in the required form.