Antisymmetric Exchange

Magnetism is due to exchange rathern than magnetic dipolar forces

The exchange interaction in a Heisenberg model is usually described by an exchange constant \(J\), where \(J < 0\) for ferromagnetic order and \(J > 0\) for antiferromagnetic order

But actually, exchange might be more generally described by a matrix, which allows not only for symmetric but also antisymmetric exchange

Let’s consider as an example a classical Heisenberg model with spin represented by vectors \(\vec{S}\) and the two spins \(\vec{S}_1\) and \(\vec{S}_2\)

$$
\hat{H}_{\mathrm{ex}} = \vec{S}_1 \hat{J} \vec{S}_2
$$

Every matrix can be decomposed into a symmetric and asymmetric part

$$
\hat{J} = \frac{\hat{J} + \hat{J}^T}{2} + \frac{\hat{J} – \hat{J}^T}{2} = \hat{J}^S + \hat{J}^A
$$

So for three-dimensional spin vectors we generally have

$$
\hat{J}^S = \begin{pmatrix} J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & J \end{pmatrix}
$$

and

$$
\hat{J}^A = \begin{pmatrix} 0 & D_{12} & D_{13} \\ -D_{12} & 0 & D_{23} \\ -D_{13} & -D_{23} & 0 \end{pmatrix}
$$

So the Hamiltonian can also be decomposed into two parts

$$
\begin{aligned}
\hat{H}_{\mathrm{ex}} =& \vec{S}_1 \hat{J}^S \vec{S}_2 + \vec{S}_1 \hat{J}^A \vec{S}_2 \\
=& J S_1^x S_2^x + J S_1^y S_2^y + J S_1^z S_2^z \\
& + S_1^x D_{12} S_2^y – S_1^y D_{12} S_2^x
+ S_1^x D_{13} S_2^z – S_1^z D_{13} S_2^x
+ S_1^y D_{23} S_2^z – S_1^z D_{23} S_2^y \\
=& J \vec{S}_1 \cdot \vec{S}_2 + \vec{D} \cdot \vec{S}_1 \times \vec{S}_2
\end{aligned}
$$

The second term is the antisymmetric exchange, since it changes sign when you swap the two spins, and it stems from the so-called Dzyaloshinskii-Moriya interaction (DMI) in crystals with broken inversion symmetry.

The Hamiltonian can only have terms that are consistent with the crystal symmetry, therefore only if inversion symmetry is broken the DMI term is allowed.

The orientation of the DMI vector \(D\) breaks inversion, as the sign of the DMI energy depends on the orientation of the spins relativ to it. This enables textures such as spin cycloids or spin helix, which have a chirality.