Non-chiral skyrmion v.s. Left/Right chiral skyrmion

Question

A skyrmion in a 3-dimensional space (or a 3-dimensional spacetime) is detected by a topological index $$n= {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial \mathbf{M}}{\partial x}\times\frac{\partial \mathbf{M}}{\partial y}\right)dx dy $$ where $M$ is the vector field in 3-dimensions. The $x$ and $y$ are coordinates on the 2-dimensional plane (say a 2-dimensional projective plane from a stereographic projection).

Naively there are non-chiral skyrmion [Fig (a)] v.s. Left/Right chiral skyrmion [Fig (b), shown the Right chiral skyrmion].

However, under the rotation $R$ about the $z$-axis and the stereographic projection $P$, the non-chiral skyrmion and Left/Right chiral skyrmion can be transformed into each other. In other words, we can also see the topological index $n= {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial \mathbf{M}}{\partial x}\times\frac{\partial \mathbf{M}}{\partial y}\right)dx dy$ for non-chiral skyrmion and Left/Right chiral skyrmion can be the same!

Question: If the topological index $n$ is NOT a good characteristic for non-chiral skyrmion v.s. Left/Right chiral skyrmion, what would be the index for such chirality? Naively, one can propose to use the winding number $$\oint \nabla \theta \cdot dl $$ on the 2D $x-y$ plane to define the chirality. However, the full space now is in 3D, so again it is likely that the winding number can be rotated away by continuous deformation (?). Do we really have good characteristics and distinctions for non-chiral skyrmion v.s. Left/Right chiral skyrmion in 3D? [For example, do we require to use the merons or instanton-number changing to see the distinctions? How precisely could we distinguish them?] Can the distinction of non-chiral skyrmion v.s. Left/Right chiral skyrmion be seen in the semi-classical sense? Or only in the full quantum theory? [Say, a Non-linear sigma model.] Or are there really distinctions after all?

Picture web-sources from Ref 1 Wiki and Ref 2.

Answer 1

Using the Cartesian coordinates $X,Y$ on the plane and spherical coordinates $\theta, \phi$ on the sphere; the stereographic projection:

$$ X+iY = \cot\frac{\theta}{2} e^{i\phi}$$

is a singular function since it transforms the north pole $\theta = 0$ of the sphere to the big circle at infinity of the plane.

An other way to see the singularity: Using the inverse stereographic projection:

$$ \hat{M} = (\frac{X}{X^2+Y^2+1}, \frac{Y}{X^2+Y^2+1}, \frac{X^2+Y^2-1}{X^2+Y^2+1})$$

The topological charge density

$$\rho_Q = \frac{1}{4 \pi} \hat{M} \cdot \big (\partial_X \hat{M} \times \partial_Y \hat{M} ) dX dY = \frac{1}{4 \pi} \frac{dXdY}{(1+X^2+Y^2)^2} = \frac{1}{4 \pi} \sin \theta d \theta d \phi$$

is the surface area element of the sphere (normalized to a unit area).

On the plane, $\rho_Q$ is an exact form:

$$\rho_Q = dA_Q$$

With:

$$A_Q = \frac{1}{2 \pi} \frac{XdY-YdX}{1+X^2+Y^2}$$

While on the sphere it is not exact, because if we write:

$$\rho_Q = -d(\cos\theta d\phi) = -d(\phi d \cos\theta)$$

Neither $\phi$ nor $d\phi$ are global functions or forms on the sphere.

If the mapping were smooth an exact form would be transformed into an exact form which necessarily has a vanishing topological charge on a compact manifold without a boundary. It is the singularity of the stereographic projection which gives rise to the non-vanishing topological charge.

The Skyrmion chirality is not topological. Left and right Skyrmions carry the same topological charge. For a particular magnet, left and right Skyrmions are separated by an energy barrier and only one of them is the minimum energy solution. The Barrier is due to the Dzyaloshinsky-Moriya term: $$\int d^2x D \hat{M}\cdot(\nabla \times\hat{M})$$ This term exists in magnets with broken spatial reversal symmetry. Rotating the Skyrmion about the $z$-axis by an arbitrary angle $\theta$ $$\hat{M}' = (M_x \cos\theta + M_y \sin\theta , -M_x \sin\theta + M_y \cos\theta , M_z)$$ does not change the topological charge, nor the leading terms in its energy functional ,such as $J (\nabla \hat{M})^2$. However, the Dzyaloshinsky-Moriya term is linear in $\sin \theta$, thus the minimum energy is obtained at $\theta = -\frac{\pi}{2}$ when the coefficient $D$ is positive and $\theta = \frac{\pi}{2}$, when it is negative. The sign depends on the sign of the spin orbit interaction of the underlying microscopic theory.

It is possible in principle to reverse a Skyrmion chirality by controlling the parameter $D$ of the Dzyaloshinsky-Moriya term.

The Skyrmion chirality cannot be associated to a topological charge, because the transformation between a left and a right handed Skyrmion is smooth. In addition, the first homotopy group $ \pi_1(S^2) = 0$ of the $2$-sphere vanishes (the sphere has no one dimensional holes), thus the chirality cannot be expressed as a one dimensional winding like the case of a vortex.

Answer 2

Just two side remarks to David's answer:

1. Topological stability and energetical stability are different conceptually. The reason why we are interested in classifying phases using topological numbers is that usually topological different configurations have energy barriers. In the skyrmion case, to destroy one skyrmion, which is equivalent to flip the down spin in the center and it will cost energy of order $\sim A$ where A is the exchange energy between nearest spins. From the point of topology, non-chiral skyrmion and chiral skyrmion are the same and among which have no energy barrier due to topology. They are all topological degenerate. But we can introduce Dzyaloshinsky-Moriya effect into the system and create energy barrier between non-chiral skyrmion and chiral skyrmion. This energy barrier is because we introduce new physics instead of topology.

2. The winding number you defined: $$N=\frac{1}{2\pi}\oint \delta \theta$$ is well defined in the skyrmion case. But it doesn't capture the chirality. For both non-chiral and chiral skyrmion, you will get the same winding number $1$. In fact, the skyrmion number can be reduced to the winding number: $$N_{skyrmion}=\text{change in polarization} \cdot N_{\text{winding number}}$$ where changing in polarization means the changing in $z$ components for spins from infinity to the center down spin. So to count the skyrmion number, we just need to count the winding number.