Skyrmions and Magnetic Textures

A magnetic texture is a spatially varying magnetic order parameter. Domains, domain walls, vortices, spirals, bubbles, merons, skyrmions, and three-dimensional singular structures are different ways for that order parameter to vary in space. Their form is set jointly by topology, exchange, anisotropy, dipolar fields, Dzyaloshinskii–Moriya coupling, applied fields, boundaries, disorder, and temperature.

Topology answers whether one smooth texture can be continuously deformed into another while respecting a declared order-parameter space and boundary condition. It does not by itself answer whether the texture is the equilibrium state, how large it is, how long it lives, or how much current moves it. Those are energetic and dynamical questions.

Required background. Magnetic Anisotropy supplies the exchange, anisotropy and magnetostatic energies, while Homotopy and Winding supplies the winding and topological-charge language.

Helpful background. Ferromagnetism provides the familiar order-parameter branch, while Spintronics supplies current-induced torques and electrical detection.

Magnetization textures and skyrmion charge

Unless stated otherwise, consider a thin magnetic layer in the oriented $xy$ plane with normal $+\hat{\mathbf z}$. A ferromagnetic texture is represented by

$$\mathbf m(x,y) = \frac{\mathbf M(x,y)}{M_s}, \qquad |\mathbf m|=1.$$

The oriented skyrmion-density convention used here is

$$q(x,y) = \frac{1}{4\pi} \mathbf m\cdot \left( \partial_x\mathbf m \times \partial_y\mathbf m \right),$$

and the total charge is

$$Q = \int q(x,y)\, \mathrm dx\,\mathrm dy.$$

Reversing the spatial orientation, exchanging $x$ and $y$, or reversing every spin changes the sign of $Q$. A paper that quotes only “$Q=1$” without its orientation and core/background convention has not fully specified the texture.

Integer quantization requires more than smooth local magnetization. If

$$\mathbf m(\mathbf r) \longrightarrow \mathbf m_\infty \quad \text{as} \quad |\mathbf r|\to\infty,$$

the plane plus its point at infinity is topologically a sphere:

$$\mathbb R^2\cup\{\infty\} \simeq S^2.$$

The texture is then a map

$$\mathbf m:S^2_{\mathrm{space}} \longrightarrow S^2_{\mathrm{spin}},$$

whose degree belongs to

$$\pi_2(S^2) = \mathbb Z.$$

An open edge or a nonuniform far field removes this particular quantization argument, so a finite-region integral need not be an integer. A vanishing amplitude or singularity can invalidate the smooth unit-vector map itself. Constant boundary conditions are not the only route to integer degree: a smooth periodic texture defines a map from a spatial torus to the spin sphere and also has integer charge per periodic cell.

The relevant real-space boundary differs among defects:

Texture	Surrounding object in real space	Typical classification	Essential boundary statement
Domain wall	two points on opposite sides	$\pi_0(\mathcal M)$	the two domains lie in disconnected minima
Vortex	closed loop around a core	$\pi_1(\mathcal M)$	the order parameter remains in the allowed manifold on the loop
Skyrmion	compactified two-dimensional plane	$\pi_2(\mathcal M)$	the far field approaches one constant value

Here $\mathcal M$ is the actual order-parameter manifold. It need not be $S^2$. An easy-plane magnet, an antiferromagnet whose microscopic state genuinely identifies opposite Néel vectors, a noncollinear magnet, and a multiferroic can have different manifolds and different defect classifications. Homotopy and Winding supplies the general mathematical language.

Top: rotating the in-plane arrows from radial to tangential changes helicity without changing $Q=-1$ for the down-core, up-background skyrmion. The arrows show the ring where $m_z=0$; the outer symbols denote the far field. Bottom: exact enclosed charges for the two radial profiles introduced below. A restricted image of a skyrmion can contain $Q(R)=-1/2$ even though its full-plane charge is $-1$; that alone does not make the full texture a meron.

Energy is not topology

For a film of thickness $t_f$ with magnetization uniform through its thickness, a representative micromagnetic energy is

$$E[\mathbf m] = t_f\int_{\Omega}\mathrm d^2r\, \big[ A \sum_{i=x,y} (\partial_i\mathbf m)^2 + f_{\mathrm D} + K_u(1-m_z^2) - \mu_0M_sH_zm_z \big] +E_d[\mathbf m].$$

Here $\Omega$ is the film footprint, $A$ has units of joules per meter, and the local densities have units of joules per cubic meter; the factor $t_f$ makes their integral an energy. The local anisotropy $K_u$ excludes magnetostatic shape energy. The Dzyaloshinskii–Moriya density $f_{\mathrm D}$ depends on crystal and interface symmetry, while $E_d$ is the full nonlocal magnetostatic energy, including the stray field outside the film. Their canonical forms and sign conventions are developed in Magnetic Anisotropy. Replacing $E_d$ by its local thin-film approximation gives $K_{\mathrm{eff}}=K_u-\mu_0M_s^2/2$; one must then omit the full $E_d$ rather than count that shape term twice.

Under a dilation

$$\mathbf m_\lambda(\mathbf r) = \mathbf m(\mathbf r/\lambda),$$

at fixed film thickness, consider the finite excess energy relative to the uniform background. For a sufficiently localized profile on the whole plane, the local terms scale as

$$\begin{aligned} E_{\mathrm{ex}}(\lambda) &\propto \lambda^0, \\ E_{\mathrm D}(\lambda) &\propto \lambda, \\ E_K(\lambda), \ E_H(\lambda) &\propto \lambda^2. \end{aligned}$$

Dipolar energy and finite boundaries have additional geometry dependence. A finite texture radius can be selected when terms with different scaling balance. Topological charge alone supplies no length scale.

For the ideal two-dimensional $O(3)$ sigma model, define

$$E_\sigma = \frac{J}{2} \int\mathrm d^2r\, \left[ (\partial_x\mathbf m)^2 + (\partial_y\mathbf m)^2 \right].$$

Here $J$ has units of energy; the film exchange term above corresponds to $J=2At_f$. Completing a square gives the Belavin–Polyakov bound

$$E_\sigma \ge 4\pi J|Q|.$$

Configurations saturating the bound have a continuous size modulus: exchange alone neither selects a preferred radius nor makes the skyrmion the thermodynamic ground state. Lattice discreteness can remove scale invariance and permit collapse, while Dzyaloshinskii–Moriya coupling, frustrated exchange, anisotropy, field, dipolar energy, confinement, or higher-order interactions can stabilize a finite size.

Three stability statements must be kept separate:

• topological distinction means a smooth deformation is obstructed under declared boundary and fixed-length assumptions;
• metastability means the texture is a local free-energy minimum separated by a finite barrier;
• thermodynamic stability means it minimizes the relevant free energy against every competing phase.

A skyrmion can be topologically nontrivial and metastable, while a skyrmion lattice is stable only within a material- and geometry-dependent phase region.

Domain walls

An easy-axis ferromagnet has two uniform minima,

$$\mathbf m = \pm\hat{\mathbf z}.$$

A $180^\circ$ wall interpolates between them. With an oriented coordinate $x$, define a one-dimensional wall charge

$$q_{\mathrm{DW}} = \frac{1}{2} \left[ m_z(-\infty) - m_z(+\infty) \right].$$

An up-to-down wall has $q_{\mathrm{DW}}=+1$ in this convention; reversing the domains reverses the sign. This charge records the boundary interpolation and is not the two-dimensional skyrmion number $Q$.

In the local thin-film approximation, with residual nonlocal stray-field energy neglected, exchange stiffness $A$ and positive effective easy-axis anisotropy $K_{\mathrm{eff}}$ give the ideal wall scale and energy per wall area

$$\Delta = \sqrt{ \frac{A}{K_{\mathrm{eff}}} }, \qquad \sigma_0 = 4\sqrt{ AK_{\mathrm{eff}} }.$$

The corresponding energy per length of wall in the film plane is $t_f\sigma_0$. Those results, including the full minimizing profile and width-convention warning, are derived at From anisotropy to magnetic textures.

A Bloch wall rotates through an in-plane direction parallel to the wall, whereas a Néel wall rotates through the wall normal. Magnetostatics, thickness, crystal symmetry, and interfaces decide which costs less. Dzyaloshinskii–Moriya coupling can select both wall type and handedness, but inferring $D$ from a wall requires the other energy terms and boundary conditions too.

Two collective coordinates often describe a nearly rigid wall:

$$X(t), \qquad \Phi(t),$$

where $X$ is wall position and $\Phi$ its internal rotation angle. Field or current can couple translation to internal precession. Above a material- and geometry-dependent Walker scale, $\Phi$ no longer remains stationary and the simple steady-velocity picture fails. Pinning, wall tilting, deformation, and nucleation frequently matter before that ideal threshold.

Walls can terminate at a sample boundary, meet other walls, or change character through a three-dimensional singularity such as a Bloch point. Their dynamics therefore depends on both local texture and global domain geometry.

Magnetic vortices and merons

In a thin disk, magnetostatic energy often favors in-plane flux closure. Far from a small core, write

$$\mathbf m_\parallel = \left( \cos\Phi, \sin\Phi \right),$$

with

$$\Phi(\varphi) = n\varphi+\gamma.$$

The integer

$$n = \frac{1}{2\pi} \oint \boldsymbol{\nabla}\Phi\cdot \mathrm d\boldsymbol{\ell}$$

is the vorticity. The phase $\gamma$ sets circulation or helicity. Near the center, the magnetization turns out of plane to avoid an exchange singularity. Its polarity is

$$p = m_z(0) = \pm1.$$

For the axisymmetric profile

$$\mathbf m= \begin{pmatrix} \sin\theta(r)\cos[n\varphi+\gamma]\\ \sin\theta(r)\sin[n\varphi+\gamma]\\ \cos\theta(r) \end{pmatrix},$$

the skyrmion charge inside a circular region is

$$Q = \frac{n}{2} \left[ \cos\theta(0) - \cos\theta(\infty) \right].$$

A magnetic vortex with in-plane far field has

$$\cos\theta(\infty)=0,$$

so

$$Q_{\mathrm{vortex}} = \frac{np}{2}.$$

It covers half the spin sphere and is therefore called a meron. The half-integer is compatible with topology because the far-field magnetization is not one constant point on $S^2$; it winds around the equator.

Vorticity, polarity, and circulation are distinct labels. Reversing the core changes $p$ and $Q$ without changing the in-plane winding. Reversing circulation can change $\gamma$ without changing $n$ or $Q$. A vortex can annihilate with an antivortex, leave through an edge, or reverse polarity through a short-lived singular core.

The same gyroforce that appears for skyrmions produces gyrotropic vortex-core motion. Confinement supplies a restoring force, yielding a low-frequency orbit whose sense depends on core polarity under standard conventions.

Skyrmions, antiskyrmions, and helicity

For the axisymmetric ansatz above, choose a down core and up background:

$$\theta(0)=\pi, \qquad \theta(\infty)=0.$$

Then

$$Q=-n.$$

With $n=+1$, this page therefore calls the texture $Q=-1$. Other sign conventions are common.

For $n=+1$, helicities $\gamma=0,\pi$ give Néel-type radial rotation, while $\gamma=\pi/2,3\pi/2$ give Bloch-type tangential rotation.

Topological charge does not determine helicity. Two textures can have the same $Q$ and different in-plane winding patterns, energies, and responses.

An antiskyrmion has the opposite winding structure and typically anisotropic helicity around its perimeter. It is favored by selected anisotropic Dzyaloshinskii–Moriya tensors rather than by the isotropic bulk or ideal interfacial forms. A skyrmionium can contain nested walls with total $Q=0$. Magnetic bubbles can have $Q=0$ or nonzero $Q$ depending on their complete vector texture. A circular image alone does not determine the topological class.

In a bulk field-polarized background, an isolated skyrmion is commonly an excitation or metastable state; confinement can instead make a single skyrmion the equilibrium configuration of a finite sample. A skyrmion crystal is a periodic magnetic phase with one or more topological objects per magnetic unit cell. Reciprocal-space Bragg peaks establish periodic order, while real-space or polarization-sensitive information is needed to determine the vector texture and charge.

Charge within an observation window

A simple $n=1$ skyrmion profile has

$$m_z(r)=\frac{r^2-a^2}{r^2+a^2}, \qquad \sin\theta(r)=\frac{2ar}{r^2+a^2}.$$

The scale $a>0$ is the radius where $m_z=0$. Integrating only to radius $R$ gives

$$Q(R)=-\frac{R^2}{R^2+a^2}.$$

At $R=a$ the image contains only half the charge; even $R=5a$ gives $Q(R)=-25/26$. In contrast, an up-core meron with $m_z(r)=e^{-r^2/a^2}$ and $n=1$ has

$$Q(R)=\frac12\left(1-e^{-R^2/a^2}\right) \longrightarrow\frac12.$$

The distinction lies in the complete far field, not whether a finite integral happens to equal a half-integer. These analytic profiles illustrate topology; they are not asserted to minimize a specific material’s full micromagnetic energy.

How magnetic interactions select texture

Different mechanisms stabilize superficially similar objects:

• bulk Dzyaloshinskii–Moriya coupling in chiral crystals commonly favors Bloch-type twists;
• interfacial Dzyaloshinskii–Moriya coupling in polar multilayers commonly favors Néel-type twists;
• anisotropic Dzyaloshinskii–Moriya coupling can favor antiskyrmions;
• dipolar energy can stabilize magnetic bubbles and flux-closure textures even without Dzyaloshinskii–Moriya coupling;
• frustrated exchange can select finite-wavevector and multi-$Q$ textures in centrosymmetric magnets;
• itinerant multi-spin or Fermi-surface-mediated interactions can stabilize short-period noncoplanar order;
• confinement and patterned boundaries can stabilize states absent in the bulk phase diagram.

Broken inversion permits particular Dzyaloshinskii–Moriya invariants; it does not guarantee a skyrmion phase. Likewise, centrosymmetry forbids conventional pairwise Dzyaloshinskii–Moriya coupling on inversion-symmetric bonds but does not forbid all skyrmions.

Useful material families include chiral B20 magnets, polar magnets, ultrathin heavy-metal/ferromagnet multilayers, frustrated centrosymmetric magnets, ferrimagnetic films, antiferromagnets, and magnetic insulators. Their characteristic sizes span atomic to micrometre scales, and their stability windows vary greatly. “Skyrmion material” is therefore a phase-diagram statement, not merely a chemical label.

Creation, annihilation, and conditional protection

In a smooth fixed-length continuum on a closed surface, $Q$ cannot change continuously. Real magnets evade one of those assumptions through:

• escape across a physical boundary;
• a Bloch point or other three-dimensional singularity;
• local suppression of $|\mathbf M|$ near a high-temperature or itinerant core;
• atomic-scale lattice evolution for which the continuum map ceases to be smooth;
• nucleation from a defect, notch, contact, or strongly driven region.

Topology can therefore raise an activation barrier and prolong lifetime, but it does not make a texture indestructible. The barrier depends on the actual annihilation path and can be much smaller at an edge or defect than in a perfect bulk calculation.

Emergent electrodynamics

When a conduction electron follows a nondegenerate local spin band adiabatically, its state accumulates a real-space Berry phase. Let $e>0$, let $\pm$ label eigenvalues $\pm1$ of $\boldsymbol\sigma\cdot\mathbf m$, and represent the effective Lorentz force using the electron charge $-e$ for both bands. This convention gives

$$B_{\mathrm{em},z}^{\pm} = \pm \frac{\hbar}{2e} \mathbf m\cdot \left( \partial_x\mathbf m \times \partial_y\mathbf m \right).$$

The emergent flux through one smooth texture is

$$\begin{aligned} \Phi_{\mathrm{em}}^\pm &= \int B_{\mathrm{em},z}^{\pm} \mathrm dx\,\mathrm dy \\ &= \pm \frac{h}{e} Q. \end{aligned}$$

The labels refer to physical spin, not automatically to majority and minority populations or magnetic-moment direction. With the conventions fixed, the magnitude is one ordinary flux quantum $h/e$ per unit $|Q|$ in this adiabatic continuum limit. It is an effective Berry field, not an additional Maxwell magnetic flux measured by a pickup coil.

For a time-dependent texture, the corresponding emergent electric field can be written

$$E_{\mathrm{em},i}^{\pm} = \pm \frac{\hbar}{2e} \mathbf m\cdot \left( \partial_i\mathbf m \times \partial_t\mathbf m \right).$$

For rigid translation, $\partial_t\mathbf m=-(\mathbf v\cdot\boldsymbol\nabla)\mathbf m$, so $\mathbf E_{\mathrm{em}}^\pm=-\mathbf v\times\mathbf B_{\mathrm{em}}^\pm$. This provides a direct sign check and explains the spin-dependent electromotive response (Everschor-Sitte and Sitte, 2014). The adiabatic approximation requires texture and time scales slow enough to suppress transitions between the local spin bands.

On a lattice, the local precursor is scalar spin chirality,

$$\chi_{ijk} = \mathbf m_i\cdot \left( \mathbf m_j\times\mathbf m_k \right).$$

For a small triangle of nearly parallel spins, the oriented spherical area is $\Omega_{ijk}\simeq\chi_{ijk}/2$. Summing those areas gives the continuum Berry flux. At atomic scales, however, band structure, nonadiabaticity, multiple orbitals, and momentum-space Berry curvature can invalidate a simple emergent-field estimate.

An often used one-band estimate is

$$\rho_{yx}^{\mathrm T} \simeq P R_0 \langle B_{\mathrm{em}}\rangle,$$

For one parabolic electron band split into two adiabatic spin channels with equal mobilities and weak Hall angles, $R_0=-1/(ne)$, $P=(n_+-n_-)/n$ and $B_{\mathrm{em}}=B_{\mathrm{em}}^+$. Unequal mobilities change the weighting of the two spin channels; $P$ cannot then be inferred from carrier numbers alone. The estimate is not a universal extraction formula. Multiband transport, anomalous Hall evolution, nonadiabatic scattering, inhomogeneity, and changing magnetic phases can all produce or reshape a Hall residual. A “topological Hall” hump is supporting evidence only when combined with independent texture information and a controlled background model.

Rigid-texture dynamics

Projecting the Landau–Lifshitz–Gilbert equation onto a slowly translating texture gives a Thiele-type force balance. Using the negative-precession-sign Gilbert equation and the texture torques defined in Spintronics, the force balance is

$$\mathbf G \times \left( \mathbf v-\mathbf u \right) + \mathsf D \left( \alpha\mathbf v - \beta\mathbf u \right) + \boldsymbol{\nabla}_{\mathbf R}U - \mathbf F_{\mathrm{ext}} = 0.$$

Here:

• $\mathbf v=\dot{\mathbf R}$ is the velocity of the texture position $\mathbf R$;
• $\mathbf u$ is the signed adiabatic spin-transfer velocity in the term $-(\mathbf u\cdot\boldsymbol\nabla)\mathbf m$, not necessarily the electron drift velocity;
• $\mathbf G=G\hat{\mathbf z}$ is the gyrovector;
• $\mathsf D$ is a symmetric positive-semidefinite dissipative tensor built from texture gradients, with material prefactors included so that each term is a force;
• $\alpha\ge0$ is Gilbert damping and $\beta$ a nonadiabatic coefficient;
• $U(\mathbf R)$ is a static potential energy, including confinement, defects, and interactions;
• $\mathbf F_{\mathrm{ext}}$ represents drives not already included in $U$, such as an applied spin–orbit torque.

For a uniform ferromagnetic film in these conventions,

$$\begin{aligned} s&=\frac{M_s t_f}{|\gamma|},\\ G&=4\pi sQ,\\ D_{ij}&=s\int \partial_i\mathbf m\cdot\partial_j\mathbf m\,\mathrm d^2r. \end{aligned}$$

Here $s$ is the positive angular-momentum magnitude per area and $|\gamma|$ is in $\mathrm{s^{-1}T^{-1}}$. The sign of $G$ follows from the choice of magnetization rather than electron-spin direction and the displayed Gilbert/force conventions; other conventions reverse it together with other signs. Projection onto $\mathbf m\times\partial_i\mathbf m$ gives the antisymmetric integral $\int\mathbf m\cdot(\partial_x\mathbf m\times\partial_y\mathbf m)\,\mathrm d^2r=4\pi Q$.

For $\mathbf u=0$, taking the scalar product with $\mathbf v$ checks the force signs:

$$\frac{\mathrm dU}{\mathrm dt} =\mathbf F_{\mathrm{ext}}\cdot\mathbf v -\alpha\,\mathbf v\cdot\mathsf D\mathbf v.$$

Without an external drive the potential energy cannot increase. The gyroscopic force does no work.

The gyroscopic term is transverse to motion and produces a skyrmion Hall angle in a ferromagnet. Damping produces a longitudinal drag. Pinning can impose a threshold. Smooth deformation changes $\mathsf D$ and the drive coupling, but at fixed $Q$, $M_s$, $t_f$ and $\gamma$ it does not change $G=4\pi sQ$. Spatially varying material weights, escape through an edge or topology change require reconsidering that statement.

In an ideal bipartite antiferromagnet, opposite sublattice gyrovectors can cancel, suppressing transverse drift. In a ferrimagnet, the gyroscopic coefficient can pass through zero near angular-momentum compensation. Neither cancellation is automatic in a real multilayer, canted state or uncompensated interface. Near compensation, inertial and internal-mode terms omitted from this first-order equation can become important.

The Thiele equation omits internal modes. Skyrmions can breathe, gyrate, deform, split, couple to magnons, and acquire effective inertia through additional coordinates. Near annihilation, collision, or strong drive, a rigid-particle interpretation can fail completely.

Computing charge from discrete data

Directly replacing derivatives by finite differences is fragile when a wall spans only a few pixels. A geometric method triangulates the image or simulation mesh. For an oriented triangle $(i,j,k)$ of unit vectors, define

$$\chi_{ijk}=\mathbf m_i\cdot(\mathbf m_j\times\mathbf m_k),$$ $$d_{ijk}={}1+\mathbf m_i\cdot\mathbf m_j +\mathbf m_j\cdot\mathbf m_k+\mathbf m_k\cdot\mathbf m_i.$$

Its signed solid angle is

$$\Omega_{ijk}=2\operatorname{atan2}(\chi_{ijk},d_{ijk}).$$

Then

$$Q_{\mathrm{lat}} = \frac{1}{4\pi} \sum_{\triangle} \Omega_{ijk}.$$

All triangles must use one orientation and the same geodesic interpolation convention (Berg and Lüscher, 1981). Antipodal pairs are ambiguous, but they are not the only exceptional configurations. If the triple product vanishes while the second argument of $\operatorname{atan2}$ is nonpositive, the branch can jump. For example, three equatorial spins separated by $120^\circ$ bound either hemisphere, despite having no antipodal pair.

Use enough spatial resolution to follow the wall, and test changes of mesh, triangulation and reconstruction. The complete vector field and its boundary treatment are essential: normalizing noisy pixels or artificially fixing an image boundary can change the inferred charge. For a nonexceptional closed or periodic triangulation, the geometric sum has integer degree; an open crop need not.

Experimental signatures

No single probe universally measures $Q$.

Probe	Primary sensitivity	What remains model dependent
Lorentz transmission electron microscopy	projected in-plane induction and deflection contrast	three-dimensional magnetization, core polarity, thickness and defocus transfer
Spin-polarized scanning tunneling microscopy	surface magnetic contrast with atomic resolution	tip polarization, vector reconstruction, subsurface texture
XMCD photoemission or transmission microscopy	element-specific magnetization projection	missing vector components, depth averaging, domain overlap
MFM or scanning NV magnetometry	stray field above the sample	inverse reconstruction, material parameters, buried layers
Small-angle neutron or resonant x-ray scattering	periodic wavevectors, orientation, selected chirality information	individual-object topology and defects
Hall transport	transverse electrical response	ordinary/anomalous backgrounds, adiabaticity, multiband coefficients
Microwave and terahertz spectroscopy	collective gyration and breathing modes	mode assignment and coexistence with non-skyrmion phases

From contrast to vector topology

Magnetic contrast first constrains candidate textures. Multiple projections, field and temperature dependence, and a forward model of the probe then help distinguish a trivial bubble from a Néel or Bloch skyrmion. A numerical charge requires enough information to determine the vector field, including its core and boundary. Scattering, transport and mode spectra provide independent constraints on the same model.

Imaging circles plus a Hall hump is not yet a topology measurement. Conversely, imperfectly quantized $Q$ from finite-resolution open-boundary data does not automatically refute a skyrmion interpretation. Compare the uncertainty from missing components, reconstruction and finite field of view before assigning a topological class. Hall Measurements discusses the transport backgrounds in more detail.

Exercises

1. Charge of an axisymmetric texture

For

$$\mathbf m= \begin{pmatrix} \sin\theta\cos[n\varphi+\gamma]\\ \sin\theta\sin[n\varphi+\gamma]\\ \cos\theta \end{pmatrix},$$

derive

$$Q = \frac{n}{2} \left[ \cos\theta(0) - \cos\theta(\infty) \right].$$

Does $Q$ depend on helicity $\gamma$?

Solution

In polar coordinates,

$$\mathrm dx\,\mathrm dy = r\,\mathrm dr\,\mathrm d\varphi,$$

and

$$\mathbf m\cdot(\partial_x\mathbf m\times\partial_y\mathbf m) =\frac1r\mathbf m\cdot (\partial_r\mathbf m\times\partial_\varphi\mathbf m).$$

For the ansatz,

$$\mathbf m\cdot \left( \partial_r\mathbf m \times \partial_\varphi\mathbf m \right) = n\theta'(r)\sin\theta.$$

Therefore

$$\begin{aligned} Q &= \frac{n}{2} \int_0^\infty \theta'\sin\theta\, \mathrm dr \\ &= \frac{n}{2} \left[ \cos\theta(0) - \cos\theta(\infty) \right]. \end{aligned}$$

The constant $\gamma$ drops out. Helicity changes the in-plane pattern but not the degree.

2. Belavin–Polyakov bound

Starting from

$$E_\sigma = \frac{J}{2} \int \left[ (\partial_x\mathbf m)^2 + (\partial_y\mathbf m)^2 \right] \mathrm d^2r,$$

use $|\mathbf m|=1$ to prove

$$E_\sigma \ge 4\pi J|Q|.$$

Solution

Because $\mathbf m\cdot\partial_i\mathbf m=0$,

$$|\mathbf m\times\partial_y\mathbf m|^2 = |\partial_y\mathbf m|^2.$$

Positivity gives

$$0 \le \frac{J}{2} \int \left| \partial_x\mathbf m \pm \mathbf m\times\partial_y\mathbf m \right|^2 \mathrm d^2r.$$

Expanding the square and using the scalar triple product yields

$$0 \le E_\sigma \mp 4\pi JQ.$$

Choose the sign appropriate to the sign of $Q$:

$$E_\sigma \ge 4\pi J|Q|.$$

Saturating the bound constrains the texture to a first-order self-duality equation. The bound fixes energy within a charge sector but does not select a radius.

3. A moving texture and Faraday induction

A rigid texture moves with constant velocity $\mathbf v=(v_x,v_y,0)$. Use the emergent-field definitions to find $\mathbf E_{\mathrm{em}}$ and check Faraday’s law for a smooth $B_{\mathrm{em},z}(x-v_xt,y-v_yt)$.

Solution

Substitution of $\partial_t\mathbf m=-v_x\partial_x\mathbf m-v_y\partial_y\mathbf m$ gives

$$\begin{aligned} E_{\mathrm{em},x}&=-v_yB_{\mathrm{em},z},\\ E_{\mathrm{em},y}&=v_xB_{\mathrm{em},z}. \end{aligned}$$

Thus $\mathbf E_{\mathrm{em}}=-\mathbf v\times\mathbf B_{\mathrm{em}}$. Its curl obeys

$$(\boldsymbol\nabla\times\mathbf E_{\mathrm{em}})_z =\mathbf v\cdot\boldsymbol\nabla B_{\mathrm{em},z} =-\partial_t B_{\mathrm{em},z}.$$

Both spin bands satisfy the relation, with opposite fields. Singularities where the smooth spin-band projection fails require separate treatment.

4. Hall angle from a simple Thiele balance

Set $\mathbf u=0$, take $U$ constant, and apply $\mathbf F_{\mathrm{ext}}=F\hat{\mathbf x}$, so that

$$G\hat{\mathbf z}\times\mathbf v + \alpha D\mathbf v = F\hat{\mathbf x},$$

with scalar $D>0$. Find $v_x$, $v_y$, and the Hall angle.

Solution

Since

$$\hat{\mathbf z}\times\mathbf v = - v_y\hat{\mathbf x} + v_x\hat{\mathbf y},$$

the component equations are

$$-Gv_y+\alpha Dv_x=F,$$

and

$$Gv_x+\alpha Dv_y=0.$$

Solving,

$$v_x = \frac{ F\alpha D }{ G^2+(\alpha D)^2 },$$

and

$$v_y = - \frac{ FG }{ G^2+(\alpha D)^2 }.$$

Thus

$$\tan\Theta_{\mathrm{SkH}} = \frac{v_y}{v_x} = - \frac{G}{\alpha D}.$$

The sign follows the stated gyrovector and force conventions. Pinning, drive-dependent deformation, and edges can change the observed angle.

5. An ambiguous triangle without antipodal spins

Take three unit vectors in the equatorial plane with azimuths $0$, $2\pi/3$ and $4\pi/3$. Evaluate the two arguments of the solid-angle formula. What changes if all three acquire the same small positive or negative $z$ component before normalization?

Solution

Every pair has dot product $-1/2$, while the triple product vanishes. The two arguments are therefore $0$ and $1-3/2=-1/2$. The triangle runs around a great circle; choosing either hemisphere gives an area of magnitude $2\pi$.

An infinitesimal positive common $z$ component gives a positive triple product, selecting $\Omega\to+2\pi$. A negative component selects $\Omega\to-2\pi$. The jump is $4\pi$, or one unit in the sum defining $Q_{\mathrm{lat}}$. No pair was antipodal. Such a configuration needs a finer spatial model or an explicit interpolation choice; rounding the resulting charge does not resolve the ambiguity.

References

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