# How to formulate the second quantization of DMI interaction?

The Dzyaloshinskii-Moriya interaction (DMI) existing in the interface of the ferromagnetic insulator and the metal with strong spin-orbit coupling (SOC) is shown below.

Mathematically, it can be written as: $$$$\vec{D}_{ij}\cdot[\vec{S}_i\times\vec{S}_j]$$$$

In this paper, the Rashba SOC has been second quantized. So my question is how can we second quantize the DMI-type SOC?

• @CodyAldaz Fixed it.
– Jack
Aug 4 at 23:34
• There are many ways to rewrite spin operators using either fermionic or bosonic creation and annihilation operators, if that's all you mean by second quantizing the DMI. Or are you after something more? Aug 5 at 1:23
• @Anyon You mean what I need to do is: replace $\vec{S}$ by creation and annihilation operators?
– Jack
Aug 5 at 1:27