The Heisenberg hamiltonian is $H= J\vec{S_i}\cdot\vec{S_j}$ whilst the Dzyaloshinkii Moriya Interaction is $H=\vec D\cdot\left(\vec{S_i}\times\vec{S_j}\right)$. But how do these equations come about? How are they derived? How is that one is a dot product and the other is a cross product?
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