Common Mathematical Form: All structures on this page are derived from a single unit vector field defined in polar coordinates ($r$, $\varphi$):

$$ \mathbf{m}(r,\varphi) = \big(\sin\theta(r)\cos\Phi,\ \sin\theta(r)\sin\Phi,\ p\cos\theta(r)\big), \qquad \Phi = m\,\varphi + \gamma $$

Meaning of the parameters:

• $\theta(r)$, profile: The polar angle of the spin measured from the core direction; at the centre $\theta(0) = 0$. Its value far away, $\theta_\infty$, sets the type of structure: $\pi$ for a skyrmion, $2\pi$ for a skyrmionium, $3\pi$ for a target skyrmion, $\pi/2$ for a meron.
• $p$, polarity: The direction of the core ($p = +1$: $+z$, $p = -1$: $-z$). As on the other pages, the default is $p = -1$: the core points along $-z$ (red) and the background along $+z$ (blue).
• $m$, vorticity: The number of turns made by the in-plane component of the spin during one turn around the centre. $m = 1$ for a skyrmion, $m = -1$ for an antiskyrmion, and $|m| \geq 2$ for a higher-order skyrmion.
• $\gamma$, helicity: The constant angle between the in-plane component and the radial direction. For $m = 1$, $\gamma = 0$ is an outward-pointing Néel skyrmion, $\gamma = \pi$ an inward-pointing Néel skyrmion, and $\gamma = \pm\pi/2$ Bloch skyrmions of the two chiralities. The Néel type is stabilized by interfacial DMI and the Bloch type by bulk DMI (see the pages with DMI).

Profile Functions: The profile is chosen as a $360^\circ$ domain wall of radius $R$ and width $w$. At $r = R$, $m_z = 0$; $w$ sets how sharp the transition is. For the $k\pi$ structures, $k$ walls are nested ($k = 1$ skyrmion, $k = 2$ skyrmionium, $k = 3$ target skyrmion); for the meron the profile extends only up to $\pi/2$:

$$ \theta_{\text{sk}}(r; R) = 2\arctan\!\left[\frac{\sinh(R/w)}{\sinh(r/w)}\right], \qquad \theta(r) = k\pi - \sum_{j=1}^{k} \theta_{\text{sk}}\!\left(r; \tfrac{jR}{k}\right), \qquad \theta_{\text{meron}}(r) = \tfrac{1}{2}\left[\pi - \theta_{\text{sk}}(r; R)\right] $$

These profiles are kinematic definitions chosen for visualization; they do not minimize any energy. In a real material the shape of the profile is set by the balance between exchange, DMI, external field and anisotropy.

Bimeron: The counterpart of a skyrmion in an in-plane magnetized medium is obtained by rotating every spin of the skyrmion field by $90^\circ$ about the $y$ axis:

$$ (m_x,\ m_y,\ m_z) \;\longrightarrow\; (m_z,\ m_y,\ -m_x) $$

The background now points along $\mp x$, and the structure consists of a meron and an antimeron with opposite $m_z$ (the blue and red lobes in the colour map). Because the same rotation is applied to all spins, the topological charge does not change.

Topological Charge: The topological charge $Q$ counts how many times the field wraps the unit sphere. For the form above the integral can be evaluated analytically:

$$ Q = \frac{1}{4\pi}\int \mathbf{m}\cdot\left(\partial_x \mathbf{m} \times \partial_y \mathbf{m}\right) dx\,dy = \frac{p\,m}{2}\left[1 - \cos\theta_\infty\right] $$

$Q$ depends only on the polarity, the vorticity and the far value of the profile; it is independent of the helicity $\gamma$, the radius $R$ and the wall width $w$. The control panel shows this analytic value together with a numerical value computed on a $161 \times 161$ grid with the Berg–Lüscher definition used on the other pages. The numerical value deviates from the analytic one when the tail of the structure extends beyond the finite drawing area: the meron gives $\pm 0.495$, and for very large $R$ and $w$ the charge of the skyrmion also moves away from $\pm 1$. This deviation shows that the topological charge is an integer only when the whole structure is considered together with the uniform background around it.

Unit Sphere and Wrapping: Every spin is a point on the unit sphere, and a spin texture is a map from the plane to the sphere. The sphere on the right shows all spins of the drawing area as dots. The coloured rings in the plane are chosen so that their images on the sphere are evenly spaced circles of latitude. Each ring is traversed counterclockwise in the plane (arrows). On the sphere the circle of the same colour is traversed, viewed from $+z$, counterclockwise if $m > 0$ and clockwise if $m < 0$. The signed area swept by a structure on the sphere is $4\pi Q$:

• The circles of a skyrmion advance from one pole to the other and cover the sphere exactly once.
• The circles of a meron stop at the equator and cover only a hemisphere.
• For a skyrmionium the circles reach the opposite pole and come back. The sphere is covered twice in opposite senses, and the net area is zero.
• For a bimeron the circles are arranged around the $x$ axis instead of the $z$ axis; the way the sphere is covered is the same as for a skyrmion.

When the cursor is moved over the plane, the image of the spin at that point is marked on the sphere.

Comparison of the Structures:

Structure	$\theta(0) \to \theta_\infty$	$m$	$Q$	On the sphere	Typical systems
Skyrmion (Néel / Bloch)	$0 \to \pi$	$1$	$p$	covers it once	Néel: multilayer thin films with interfacial DMI; Bloch: chiral B20 compounds (MnSi, FeGe)
Antiskyrmion	$0 \to \pi$	$-1$	$-p$	once, in the opposite sense	Heusler compounds with $D_{2d}$ symmetry
Higher-order skyrmion	$0 \to \pi$	$2, 3, \dots$	$m\,p$	$|m|$ times	Predicted theoretically for frustrated magnets
Skyrmionium	$0 \to 2\pi$	$1$	$0$	twice, in opposite senses	Magnetic thin films and nanodisks
Target skyrmion ($3\pi$)	$0 \to 3\pi$	$1$	$p$	three times, net once	Confined geometries (nanodisks)
Meron	$0 \to \pi/2$	$1$	$p/2$	a hemisphere	In-plane magnets (vortex cores); meron–antimeron lattices in chiral magnets
Antimeron	$0 \to \pi/2$	$-1$	$-p/2$	a hemisphere, in the opposite sense	Together with merons
Bimeron	skyrmion rotated by $90^\circ$	$1$	$p$	once (around the $x$ axis)	In-plane magnetized films

Suggested Experiments:

1. Helicity and topology: Select a skyrmion and press "Rotate helicity". The structure changes continuously from Néel to Bloch and back to Néel, but $Q$ does not change. This transformation is a continuous deformation.
2. Polarity: Press "Reverse polarity": the sign of $Q$ changes. In the plane this cannot be achieved by a continuous deformation; the spin at the core has to flip abruptly.
3. Antiskyrmion: Switch between skyrmion and antiskyrmion and compare the arrow directions of the circles on the sphere.
4. Skyrmionium: Observe that the circles go to the opposite pole and come back, and that the charge is zero.
5. Meron: Note that the circles stop at the equator and that $Q = \pm 1/2$.
6. Finite area: For the skyrmion, set $R$ and $w$ to their largest values. As the tail extends beyond the drawing area, the numerical $Q$ moves away from an integer.

Visualization and Color Coding: The color coding is the same as on the other pages:

• Z-Component ($S_z$): Represented by the background color of the cell. Positive values ($S_z \to +1$) transition to Dark Blue, while negative values ($S_z \to -1$) transition to Red. Intermediate values close to the $x-y$ plane approach white.
• X-Y Plane Projection: Represented by the black arrow inside each cell. The orientation and scalar length of the arrow directly indicate the $(S_x, S_y)$ vector components.
• Schematic views: The two views in the lower row show the change of orientation with large arrows. In both, the central spin points along the core direction, and on each ring the spin approaches its far value ($\theta_\infty$) in equal steps. In the 3D view the spins are drawn as thick arrows coloured by $S_z$, seen from an oblique angle; the view can be rotated by dragging. In the top view the spins are drawn as disks coloured by $S_z$ with thick arrows for the in-plane component; spins that are almost entirely out of plane are shown as ⊙ ($+z$, out of the screen) or ⊗ ($-z$, into the screen) (4 rings for the skyrmion, meron and skyrmionium, 6 for the target skyrmion). These two views are not to scale: they do not depend on $R$ and $w$, and change only with the type of structure, $p$, $m$ and $\gamma$.
• Sphere: The dots use the same color coding. Dots and circles on the back of the sphere are drawn faintly.

Limitations: This page shows only the geometry and topology of the structures; it contains no energy or dynamics. Under which conditions the structures are stable and how they move are studied on the other pages (the Monte Carlo and LLG simulations with DMI). Because the energy of an isolated meron grows logarithmically with system size, merons occur in practice in pairs or in confined geometries.