The Dzyaloshinskii-Moriya Interaction

We now examine the SOC in a spin dimer made up of two spin sites I and 2, for which the SOC Hamiltonian is given by [23] 

Suppose that 5L and 8L2 are the unquenched orbital angular momenta at sites I and 2, respectively. Then, use of the Hso (equation 24.60) as perturbation leads to the DM interaction energy Edm [23,50] 

In equation 24.61, the DM vector D12 is related to the difference in the unquenched orbital angular momenta on the two magnetic sites I and 2, namely, 

Dx SiySi - 5iz52y) + Dy S zSix — SixSiz) + D SixSiy — SiySix) 

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Sergienko, I.A. and Dagotto, E. (2006) Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B, 73, 094434. 

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