Collinear Spin-Sinusoidal Texture

• Collinear spin-sinusoidal texture is defined by a single spin component varying sinusoidally—either in momentum space for VOClBr or in real space for RMnO3.
• The analysis shows that symmetry operations and ferroelectric switching govern a robust sine-wave spin modulation, influencing magnetoelectric coupling and spintronic applications.
• Strain tuning and chemical substitution modulate the amplitude and phase of the sinusoid, decisively altering the magnetic order and enabling phase transitions between sinusoidal and antiferromagnetic states.

Searching arXiv for recent and foundational papers on collinear spin-sinusoidal textures, including VOClBr and RMnO3. Collinear spin-sinusoidal texture denotes a class of spin configurations in which the spin direction remains strictly collinear while the spin amplitude, sign, or momentum-resolved spin polarization varies sinusoidally. In current arXiv literature, the expression appears in two technically distinct settings. In monolayer Janus VOClBr, it refers to a momentum-space texture in a two-dimensional ferroelectric altermagnet, where the out-of-plane spin projection follows a sine-like dependence on $k_y$ and reverses under ferroelectric switching (Yao et al., 27 Jan 2026). In multiferroic manganites such as $R$MnO$_3$, it denotes a real-space collinear spin-density wave in which the ordered Mn moment is modulated sinusoidally along the crystallographic $b$ axis, with consequences for anomalous magnetoelectric coupling and electromagnon activity (Stenberg et al., 2011). The same literature also documents how chemical substitution can suppress such a sinusoidal phase in favor of a collinear A-type antiferromagnet (Agarwal et al., 2020).

1. Definitions and principal realizations

The defining feature is collinearity: only one spin component remains nonzero, but that component acquires a sinusoidal dependence on either crystal momentum or real-space coordinate. The resulting texture is therefore distinct from a cycloid or spiral, where the spin orientation itself rotates.

Setting	Representative system	Defining form
Momentum-space spin texture	Monolayer Janus VOClBr	$S_z(k)\propto \sin(k_y a)$
Real-space spin-density wave	Pure TbMnO$_3$ in the collinear phase	$m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$
Collinear endpoint after suppression of the sinusoid	Tb$_{0.6}$Pr$_{0.4}$MnO$_3$	$R$0 collinear A-type AFM

In VOClBr, the sinusoidal dependence is a momentum-space consequence of magnetic-crystal symmetry combined with broken inversion, and the texture is described in the source paper as a collinear spin-sinusoidal, or “d-wave,” pattern (Yao et al., 27 Jan 2026). In TbMnO$R$1, the sinusoid is a one-dimensional real-space modulation of a moment aligned along $R$2, preceding the lower-temperature cycloidal phase (Agarwal et al., 2020). In Tb$R$3Pr$R$4MnO$R$5, the incommensurate sinusoidal precursor is absent, and the system orders directly into a collinear A-type antiferromagnet (Agarwal et al., 2020).

2. Symmetry origin in monolayer VOClBr

In the distorted ferroelectric-altermagnetic phase of monolayer VOClBr, the magnetic space group contains a twofold rotation about $R$6, $R$7, under which $R$8 and $R$9, and a mirror about $_3$0, $_3$1, under which $_3$2 and $_3$3. Combined with broken inversion, these operations force any spin-splitting term $_3$4 to be odd in $_3$5 and even in $_3$6 (Yao et al., 27 Jan 2026).

Near $_3$7, the resulting two-band model is

$_3$8

with

$_3$9

Here $b$0 is the V–V lattice constant, $b$1 is an effective mass, and $b$2 is the Pauli matrix for spin. The out-of-plane spin projection on a Bloch state $b$3 is

$b$4

far from band crossings. Around the $b$5 valley, an analogous symmetry-allowed term appears,

$b$6

so that $b$7 to leading order (Yao et al., 27 Jan 2026).

This construction fixes the essential character of the texture. The spin polarization is purely out of plane, its sign changes under $b$8, and its phase zeros are symmetry-determined rather than accidental.

3. First-principles manifestation and ferroelectric reversal

The DFT+$b$9 band structures reported for VOClBr show a clear spin splitting only along the S–I–S′ path, which is parallel to $S_z(k)\propto \sin(k_y a)$0. Extracting $S_z(k)\propto \sin(k_y a)$1 and $S_z(k)\propto \sin(k_y a)$2 yields

$S_z(k)\propto \sin(k_y a)$3

with a best-fit $S_z(k)\propto \sin(k_y a)$4–$S_z(k)\propto \sin(k_y a)$5 (Yao et al., 27 Jan 2026). The computed spin polarization,

$S_z(k)\propto \sin(k_y a)$6

is excellently fitted by

$S_z(k)\propto \sin(k_y a)$7

with $S_z(k)\propto \sin(k_y a)$8–$S_z(k)\propto \sin(k_y a)$9.

Ferroelectric switching changes the sign of this entire sinusoid. The ferroelectric polarization $_3$0 along $_3$1 breaks inversion so that $_3$2, while inverting $_3$3 through V off-centering gives $_3$4. The low-energy Hamiltonian in the two ferroelectric states is therefore

$_3$5

Physically, reversing $_3$6 swaps the local Cl/Br environment under V, changes the sense of the Peierls/V–V dimerization along $_3$7, and causes $_3$8 to map $_3$9. The direct consequence is $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$0: the sine-wave-shaped spin texture flips sign over the entire Brillouin zone (Yao et al., 27 Jan 2026).

The computed spin-texture plots make this explicit. For $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$1, the Brillouin-zone map of $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$2 contains red lobes with $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$3 at $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$4 and blue lobes with $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$5 at $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$6, forming a d-wave-like pattern, while a high-symmetry cut along S–I–S′ lies almost exactly on the analytic $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$7 curve. After ferroelectric reversal, the same map shows the sign-inverted lobes. The abstract further identifies robust magnetoelectric coupling, evidenced by a complete reversal of momentum-space spin polarization upon ferroelectric switching and supported by spin texture analysis and the magneto-optical Kerr effect (Yao et al., 27 Jan 2026).

4. Strain tuning, phase locking, and functional consequences

The VOClBr work also formulates an explicit strain dependence for the splitting amplitude:

$m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$8

with $m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\,\hat e_b$9 per % from DFT fits (Yao et al., 27 Jan 2026). Under biaxial compression, the V–V spacing along $_{0.6}$0 changes, the Peierls distortion is strengthened, and the sinusoidal spin contrast is amplified. Specifically, $_{0.6}$1 increases $_{0.6}$2 by about $_{0.6}$3. Under tensile strain, the same coupling is weakened; at $_{0.6}$4, $_{0.6}$5, the spin splitting is quenched, and a conventional FE-AFM with $_{0.6}$6 is restored (Yao et al., 27 Jan 2026).

The phase of the sinusoid remains fixed throughout this tuning. Its zero crossings at $_{0.6}$7 are protected by $_{0.6}$8 and $_{0.6}$9 symmetry, so strain modifies amplitude rather than phase. The abstract adds a second functional effect of compression: biaxial compression strain of $_{0.4}$0 reduces the ferroelectric polarization switching barrier by approximately $_{0.4}$1, while a tensile strain of $_{0.4}$2 induces a phase transition to an antiferromagnet (Yao et al., 27 Jan 2026).

Because the electrically controlled spin texture is locked to the magneto-optical Kerr effect signal, the source paper proposes a non-volatile, polymorphic spintronic memory device with all-electrical writing and optical readout. A plausible implication is that the collinear spin-sinusoidal texture is not merely a band-structure signature but an electrically addressable order parameter in a two-dimensional ferroic platform (Yao et al., 27 Jan 2026).

5. Real-space sinusoidal order and anomalous magnetoelectric coupling in $_{0.4}$3MnO$_{0.4}$4

In $_{0.4}$5MnO$_{0.4}$6, the collinear sinusoidal phase is formulated in real space rather than momentum space. The full model Hamiltonian is split as

$_{0.4}$7

where $_{0.4}$8 contains exchange and single-ion anisotropy, $_{0.4}$9 is a polar optical phonon, and $_3$0, $_3$1 are anomalous spin-symmetric magnetoelectric couplings (Stenberg et al., 2011). The spin Hamiltonian includes $_3$2 for nearest-neighbor ferromagnetic exchange in the $_3$3 plane, $_3$4 for next-nearest-neighbor exchange along $_3$5, $_3$6 for interlayer coupling, and $_3$7 favoring alignment along $_3$8.

Between $_3$9 and $R$00, the collinear ground state is

$R$01

Only the $R$02 component is present, so the moment is strictly collinear. Below $R$03, the system enters a cycloidal phase,

$R$04

which breaks inversion and produces a uniform ferroelectric $R$05 (Stenberg et al., 2011).

The crucial point is that in the collinear phase only the second anomalous coupling, $R$06, remains active. Minimizing $R$07 with respect to the ionic displacement yields an incommensurate oscillatory polarization of wavevector $R$08 along $R$09,

$R$10

where $R$11. Because $R$12 couples different spin components, it breaks the residual $R$13 rotational invariance about $R$14 and pins a static oscillatory polarization at $R$15 (Stenberg et al., 2011).

This same coupling determines the spectroscopy of the phase. Linearization around the collinear state gives four magnon branches, two cyclons and two extra-cyclons, but only one cyclon is dipole-active in the collinear phase. Its dynamical equation is centered at $R$16, so only that mode hybridizes with the $R$17-axis phonon. The dielectric response therefore contains a single electromagnon Lorentz oscillator. Experimentally, the collinear phase shows one low-energy electromagnon for light polarized $R$18, while the higher-energy zone-edge electromagnon present in the cycloid disappears. X-ray diffraction simultaneously detects an oxygen-displacement modulation at wavevector $R$19, in agreement with the predicted incommensurate oscillatory polarization (Stenberg et al., 2011).

6. Suppression of the sinusoidal phase in Tb$R$20Pr$R$21MnO$R$22

Neutron powder diffraction on Tb$R$23Pr$R$24MnO$R$25 shows how a material can evolve away from the sinusoidal regime into a conventional collinear antiferromagnet. In pure TbMnO$R$26, the Mn sublattice orders at $R$27 with an incommensurate propagation vector $R$28 and this wavevector locks in to $R$29 at $R$30. The corresponding sinusoidal modulation is

$R$31

with $R$32 at $R$33, $R$34, and direction $R$35 in the Pbnm setting. Below $R$36, the pure sinusoid transforms into a cycloidal spiral, with an additional component $R$37 along $R$38, breaking inversion symmetry and inducing ferroelectricity (Agarwal et al., 2020).

By contrast, in Tb$R$39Pr$R$40MnO$R$41 all magnetic peaks index with $R$42 below the ordering temperature, and the Mn sublattice orders at $R$43 directly into collinear A-type AFM. No intermediate incommensurate phase is observed. The best fit of the Mn-only order at $R$44 and $R$45 is basis vector $R$46 of the $R$47 representation, corresponding to all Mn moments parallel to $R$48 and alternating sign along $R$49 and $R$50. At $R$51, the Tb/Pr moments add a ferromagnetic component along $R$52 ($R$53), and the resulting magnetic space group is $R$54 (Agarwal et al., 2020).

The structural trend accompanying this change is quantified by the average Mn–O–Mn angle, Jahn–Teller distortion, and one-electron bandwidth $R$55. TbMnO$R$56 has $R$57, $R$58, and $R$59; Tb$R$60Pr$R$61MnO$R$62 has $R$63, $R$64, and $R$65; PrMnO$R$66 has $R$67, $R$68, and $R$69. In the minimal Hamiltonian

$R$70

the in-plane ferromagnetic exchange $R$71 scales roughly with $R$72, while $R$73 remains antiferromagnetic. As the bond angle increases and $R$74 grows, the balance shifts from the frustrated spiral regime of TbMnO$R$75 to the robust A-type regime of PrMnO$R$76; at $R$77 Pr, the system already lies on the A-type side of the phase boundary (Agarwal et al., 2020).

7. Unifying interpretation and recurrent misconceptions

The cited literature supports a precise distinction between three notions that are often conflated. First, a collinear spin-sinusoidal texture need not be a real-space spin-density wave: in VOClBr it is a momentum-space texture with $R$78, whereas in TbMnO$R$79 it is a real-space modulation with $R$80 (Yao et al., 27 Jan 2026). Second, “sinusoidal” does not imply non-collinearity: in both cases only one spin component is present, and the modulation affects sign or amplitude rather than spin orientation itself (Stenberg et al., 2011). Third, a sinusoidal phase is not equivalent to a cycloid. In $R$81MnO$R$82, the cycloid activates additional magnetoelectric couplings and supports two strong electromagnons, whereas the collinear sinusoidal phase leaves only one surviving electromagnon and coexists with an incommensurate oscillatory polarization at $R$83 (Stenberg et al., 2011).

Across these materials, the common principle is that symmetry constrains which spin component may vary and how it may vary. In VOClBr, broken inversion together with $R$84 and $R$85 enforces an out-of-plane splitting odd in $R$86, producing the momentum-space sine law and allowing ferroelectric sign control (Yao et al., 27 Jan 2026). In manganites, exchange frustration and anisotropy stabilize a one-dimensional collinear modulation, while anomalous spin-symmetric magnetoelectric coupling converts that modulation into a lattice-polarization response and a sharply restricted electromagnon selection rule (Stenberg et al., 2011). The transition from TbMnO$R$87 to Tb$R$88Pr$R$89MnO$R$90 shows the converse process: when the exchange balance changes, the sinusoidal state can be eliminated altogether in favor of a commensurate collinear A-type antiferromagnet (Agarwal et al., 2020).