---
title: Collinear Spin-Sinusoidal Texture
url: https://www.emergentmind.com/topics/collinear-spin-sinusoidal-texture
type: topic
---

# Collinear Spin-Sinusoidal Texture

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 [2601.19560]. 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 [1109.3965]. The same literature also documents how chemical substitution can suppress such a sinusoidal phase in favor of a collinear A-type antiferromagnet [2011.12553].

## 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$_3$ | $q=(0,0,0)$ 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 [2601.19560]. In TbMnO$_3$, the sinusoid is a one-dimensional real-space modulation of a moment aligned along $\hat b$, preceding the lower-temperature cycloidal phase [2011.12553]. In Tb$_{0.6}$Pr$_{0.4}$MnO$_3$, the incommensurate sinusoidal precursor is absent, and the system orders directly into a collinear A-type antiferromagnet [2011.12553].

## 2. Symmetry origin in monolayer VOClBr

In the distorted ferroelectric-altermagnetic phase of monolayer VOClBr, the magnetic space group contains a twofold rotation about $x$, $C_{2\parallel x}$, under which $(k_x,k_y)\to(k_x,-k_y)$ and $\sigma_z\to-\sigma_z$, and a mirror about $y$, $M_y$, under which $(k_x,k_y)\to(-k_x,k_y)$ and $\sigma_z\to+\sigma_z$. Combined with broken inversion, these operations force any spin-splitting term $d_z(k)\sigma_z$ to be odd in $k_y$ and even in $k_x$ [2601.19560].

Near $\Gamma$, the resulting two-band model is
$$
H_\Gamma(k)=\epsilon_0(k)\,1+d_z(k)\,\sigma_z,
$$
with
$$
\epsilon_0(k)=\frac{\hbar^2(k_x^2+k_y^2)}{2m^*},\qquad
d_z(k)=\alpha\,\sin(k_y a)+\beta\,k_xk_y.
$$
Here $a\approx 3.4\,\text{\AA}$ is the V–V lattice constant, $m^*$ is an effective mass, and $\sigma_z$ is the Pauli matrix for spin. The out-of-plane spin projection on a Bloch state $|\psi_{n,k}\rangle$ is
$$
S_z^{(n)}(k)\equiv \langle\psi_{n,k}|\sigma_z|\psi_{n,k}\rangle
=\frac{d_z(k)}{\sqrt{d_z(k)^2+\cdots}}
\simeq \frac{\alpha\sin(k_y a)}{|\alpha\sin(k_y a)|}
\propto \sin(k_y a),
$$
far from band crossings. Around the $K$ valley, an analogous symmetry-allowed term appears,
$$
H_K(q)=\epsilon_K+v_x q_x\sigma_x+v_y q_y\sigma_y+[\lambda_1\sin(q_y a')+\lambda_2 q_x]\sigma_z,
$$
so that $S_z(q)\propto \sin(q_y a')$ to leading order [2601.19560].

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

## 3. First-principles manifestation and ferroelectric reversal

The DFT+$U$ band structures reported for VOClBr show a clear spin splitting only along the S–I–S′ path, which is parallel to $k_y$. Extracting $E_\uparrow(k_y)$ and $E_\downarrow(k_y)$ yields
$$
\Delta E(k_y)\equiv E_\uparrow-E_\downarrow \simeq 2\alpha\sin(k_y a),
$$
with a best-fit $\alpha\simeq 20$–$25\,\text{meV}$ [2601.19560]. The computed spin polarization,
$$
P_s(k_y)\equiv \frac{n_\uparrow(k_y)-n_\downarrow(k_y)}{n_\uparrow+n_\downarrow},
$$
is excellently fitted by
$$
P_s(k_y)=P_0\sin(k_y a),
$$
with $P_0\approx 0.8$–$0.9$.

Ferroelectric switching changes the sign of this entire sinusoid. The ferroelectric polarization $P$ along $+x$ breaks inversion so that $\alpha(P>0)=+\alpha_0$, while inverting $P$ through V off-centering gives $\alpha(P<0)=-\alpha_0$. The low-energy Hamiltonian in the two ferroelectric states is therefore
$$
H(k;P)=\epsilon_0(k)+[\mathrm{sgn}(P)\cdot \alpha_0\sin(k_y a)+\cdots]\sigma_z.
$$
Physically, reversing $P$ swaps the local Cl/Br environment under V, changes the sense of the Peierls/V–V dimerization along $y$, and causes $C_{2x}$ to map $\sigma_z\to-\sigma_z$. The direct consequence is $S_z(k)\to -S_z(k)$: the sine-wave-shaped spin texture flips sign over the entire Brillouin zone [2601.19560].

The computed spin-texture plots make this explicit. For $P>0$, the Brillouin-zone map of $S_z(k)$ contains red lobes with $S_z>0$ at $+k_y$ and blue lobes with $S_z<0$ at $-k_y$, forming a d-wave-like pattern, while a high-symmetry cut along S–I–S′ lies almost exactly on the analytic $\sin(k_y a)$ 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 [2601.19560].

## 4. Strain tuning, phase locking, and functional consequences

The VOClBr work also formulates an explicit strain dependence for the splitting amplitude:
$$
\alpha(\epsilon)=\alpha_0[1-\gamma\,\epsilon],
$$
with $\gamma\approx 0.15$ per % from DFT fits [2601.19560]. Under biaxial compression, the V–V spacing along $y$ changes, the Peierls distortion is strengthened, and the sinusoidal spin contrast is amplified. Specifically, $\epsilon=-4\%$ increases $\alpha$ by about $60\%$. Under tensile strain, the same coupling is weakened; at $\epsilon\approx +3\%$, $\alpha\to 0$, the spin splitting is quenched, and a conventional FE-AFM with $S_z(k)\simeq 0$ is restored [2601.19560].

The phase of the sinusoid remains fixed throughout this tuning. Its zero crossings at $k_y a=0,\pi$ are protected by $M_y$ and $C_{2x}$ symmetry, so strain modifies amplitude rather than phase. The abstract adds a second functional effect of compression: biaxial compression strain of $-4\%$ reduces the ferroelectric polarization switching barrier by approximately $87\%$, while a tensile strain of $+3\%$ induces a phase transition to an antiferromagnet [2601.19560].

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 [2601.19560].

## 5. Real-space sinusoidal order and anomalous magnetoelectric coupling in $R$MnO$_3$

In $R$MnO$_3$, the collinear sinusoidal phase is formulated in real space rather than momentum space. The full model Hamiltonian is split as
$$
H=H_S+H_{ph}+H_{me}^{(1)}+H_{me}^{(2)},
$$
where $H_S$ contains exchange and single-ion anisotropy, $H_{ph}$ is a polar optical phonon, and $H_{me}^{(1)}$, $H_{me}^{(2)}$ are anomalous spin-symmetric magnetoelectric couplings [1109.3965]. The spin Hamiltonian includes $J_0<0$ for nearest-neighbor ferromagnetic exchange in the $ab$ plane, $J_{2b}>0$ for next-nearest-neighbor exchange along $b$, $J_c>0$ for interlayer coupling, and $D_a,D_b>0$ favoring alignment along $\hat b$.

Between $T_{N1}$ and $T_{N2}$, the collinear ground state is
$$
\mathbf S_0(\mathbf R_n,T)=\pm S(T)\cos(Qbn+\phi)\,\hat{\mathbf b},
\qquad
\cos\Bigl(\frac{Qb}{2}\Bigr)=-\frac{J_0}{2J_{2b}}.
$$
Only the $b$ component is present, so the moment is strictly collinear. Below $T_{N2}$, the system enters a cycloidal phase,
$$
\mathbf S_0(\mathbf R_n)=S[\cos(Qbn)\,\hat{\mathbf b}+\sin(Qbn)\,\hat{\mathbf c}],
$$
which breaks inversion and produces a uniform ferroelectric $P_a\propto \sin(Qb)$ [1109.3965].

The crucial point is that in the collinear phase only the second anomalous coupling, $H_{me}^{(2)}$, remains active. Minimizing $H_{ph}+H_{me}^{(2)}$ with respect to the ionic displacement yields an incommensurate oscillatory polarization of wavevector $2Q$ along $\hat a$,
$$
\frac{e^*\,\mathbf x_n}{v_0}
=
4\,\chi_0\,S(T)^2\,
\sin\!\Bigl(\frac{Qb}{2}\Bigr)\,
g_b\,\sin[(2n+1)Qb]\;\hat{\mathbf a},
$$
where $\chi_0=e^{*2}/(m^*v_0\omega_0^2)$. Because $H_{me}^{(2)}$ couples different spin components, it breaks the residual $U(1)$ rotational invariance about $\hat b$ and pins a static oscillatory polarization at $2Q$ [1109.3965].

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 $q=k_0-2Q$, so only that mode hybridizes with the $a$-axis phonon. The dielectric response therefore contains a single electromagnon Lorentz oscillator. Experimentally, the collinear phase shows one low-energy electromagnon for light polarized $E\parallel \hat a$, while the higher-energy zone-edge electromagnon present in the cycloid disappears. X-ray diffraction simultaneously detects an oxygen-displacement modulation at wavevector $2Q$, in agreement with the predicted incommensurate oscillatory polarization [1109.3965].

## 6. Suppression of the sinusoidal phase in Tb$_{0.6}$Pr$_{0.4}$MnO$_3$

Neutron powder diffraction on Tb$_{0.6}$Pr$_{0.4}$MnO$_3$ shows how a material can evolve away from the sinusoidal regime into a conventional collinear antiferromagnet. In pure TbMnO$_3$, the Mn sublattice orders at $T_N\approx 41\,\text{K}$ with an incommensurate propagation vector $q=(0,0.27,0)$ and this wavevector locks in to $(0,0.28,0)$ at $T_{\text{lock}}\approx 26\,\text{K}$. The corresponding sinusoidal modulation is
$$
m_i=m_0\sin[2\pi\,q\!\cdot\! r_i+\phi]\;\hat e_b,
$$
with $m_0\approx 2.7\,\mu_B/\text{Mn}$ at $30\,\text{K}$, $\phi\approx 0$, and direction $\hat e_b$ in the Pbnm setting. Below $T_{\text{lock}}\approx 26\,\text{K}$, the pure sinusoid transforms into a cycloidal spiral, with an additional component $m_a\approx 0.3\,\mu_B$ along $a$, breaking inversion symmetry and inducing ferroelectricity [2011.12553].

By contrast, in Tb$_{0.6}$Pr$_{0.4}$MnO$_3$ all magnetic peaks index with $q=(0,0,0)$ below the ordering temperature, and the Mn sublattice orders at $T_N\approx 100\,\text{K}$ directly into collinear A-type AFM. No intermediate incommensurate phase is observed. The best fit of the Mn-only order at $50\,\text{K}$ and $25\,\text{K}$ is basis vector $C_x$ of the $\Gamma_5$ representation, corresponding to all Mn moments parallel to $+a$ and alternating sign along $b$ and $c$. At $1.5\,\text{K}$, the Tb/Pr moments add a ferromagnetic component along $b$ ($F_y$), and the resulting magnetic space group is $Pn'ma'$ [2011.12553].

The structural trend accompanying this change is quantified by the average Mn–O–Mn angle, Jahn–Teller distortion, and one-electron bandwidth $W$. TbMnO$_3$ has $\langle\text{Mn–O–Mn}\rangle=145.2^\circ$, $\Delta_{JT}=0.0051\,\text{\AA}$, and $W=0.081\,\text{eV}$; Tb$_{0.6}$Pr$_{0.4}$MnO$_3$ has $149.9^\circ$, $0.0028\,\text{\AA}$, and $0.084\,\text{eV}$; PrMnO$_3$ has $156.8^\circ$, $0.0018\,\text{\AA}$, and $0.090\,\text{eV}$. In the minimal Hamiltonian
$$
H=\sum_{\langle i,j\rangle}J_{ij}\,S_i\!\cdot\!S_j + D\sum_i (S_i^z)^2,
$$
the in-plane ferromagnetic exchange $J_{ab}$ scales roughly with $+W$, while $J_c$ remains antiferromagnetic. As the bond angle increases and $W$ grows, the balance shifts from the frustrated spiral regime of TbMnO$_3$ to the robust A-type regime of PrMnO$_3$; at $40\%$ Pr, the system already lies on the A-type side of the phase boundary [2011.12553].

## 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 $S_z(k)\propto \sin(k_y a)$, whereas in TbMnO$_3$ it is a real-space modulation with $\mathbf S_0(\mathbf R_n,T)=\pm S(T)\cos(Qbn+\phi)\,\hat{\mathbf b}$ [2601.19560]. 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 [1109.3965]. Third, a sinusoidal phase is not equivalent to a cycloid. In $R$MnO$_3$, 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 $2Q$ [1109.3965].

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 $C_{2\parallel x}$ and $M_y$ enforces an out-of-plane splitting odd in $k_y$, producing the momentum-space sine law and allowing ferroelectric sign control [2601.19560]. 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 [1109.3965]. The transition from TbMnO$_3$ to Tb$_{0.6}$Pr$_{0.4}$MnO$_3$ 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 [2011.12553].

Source: https://www.emergentmind.com/topics/collinear-spin-sinusoidal-texture