content_image/310897.jpg content_image/310898.jpg content_image/310896.jpg
or the long wavelength limit ( − → q −→ 0), we found 

where ̂ h is the helix axis unit vector. We have assumed that q$_{z}$ = 0 since our calculation was performed on xy-plane. If one recognizes that the true meaning of ̂ e$_{12}$ is the spatial direction alone which the spins propagate, i.e., − → q , he can see how the form − → P ∼ $^{eI}$̂ e$_{12}$ × $^{(}$̂ e$_{1}$ × ̂ e$_{2}$ ) given by KNB can be transformed in the presence of magnetic orders. 

## 0.0.5 5. Discussion 

In order to see how magnetic orders and electric order are coupled, let us go back to the original work of Moriya[19]. He derived the following expression: 

where 

<img src='content_image/310897.jpg'>

Here, J ( n, n $^{′}$, m, m$^{′}$) is the exchange interaction strength and − → l i denotes the angular momentum of the electron at site i . 

In our starting Hamiltonian, the exchange interaction comes from the charge transfer energy ε$_{p}$ − ε$_{d}$ and hybridization energy V [27]. Combined with the spin- orbit interaction in eq. (4b), DM interaction is clearly present in the system we considered. We can recast the wave functions we got previously in a form similar to Moriya’s by treating the hybridization energy and spin-orbit interaction as 

<img src='content_image/310898.jpg'>

<img src='content_image/310896.jpg'>

