into eq. (24), we found that The connection between the electric polarization and spin current can now be established. A common definition of the spin current is The results are same as those in eqs. (25). Introducing the SU(2) vector poten- tial[28] A i j = − ε$_{ijk}$ ℏ E$_{k}$/ 4 m$_{e}$c 2 where m$_{e}$ is the electron mass, the DM interac- tion can be rewritten as where is the guage field coupled to the spin current. It comes from the term − → E · ( − → p × − → σ ) in Pauli’s equation. See for example, ref. 28. The polarization results from the 15 ˜ j j i = t 4 i ℏ ∑ N ( d + R $_{N}$+ $_{a}$σ$_{j}$ d R N − d + R $_{N}$σ$_{j}$ d R $_{N}$+ a ) − → a | i (27) where − → a is the lattice vector of length a$_{0}$ in the direction of i and in our case t ≈ V $^{2}$/ | ε$_{dp}$ | . Note that the electrons hop along the direction of − → a . Equation (27) can be derived by discretizing the form i ℏ [ ψ $^{+}$σ$_{j}$ ∂$_{i}$ψ − ( ∂$_{i}$ψ $^{+}$) σ$_{j}$ψ ] / 4 m . The form in eq. (27) manifests itself in helical spin configuration. We calculated its expectation value with the state in (2) and found that 〈 ˜ j y x 〉 = ta$_{0}$ 2 ℏ cos φ sin( q$_{x}$a$_{0}$/ 2) = − ta$_{0}$ 2 ℏ ( − → s N × − → s N − $_{1}$) | $_{y}$, (28a) j x y = − ta$_{0}$ 2 ℏ sin φ sin( q$_{y}$a$_{0}$/ 2) = − ta$_{0}$ 2 ℏ ( − → s N × − → s N − $_{1}$) | $_{x}$. (28b) H$_{DM}$ = e ∑ A i j ˜ j j $_{i}$. (29) A i j = − 2 Jλ eta$_{0}$ Δ E$_{cf}$ e$_{B,i}$l$_{j}$, (30) P − → q ,k = − 2 e ∑ n,m,l,j λ 〈 m | H$_{V,i}$ | n 〉 ( ε$_{d}$ − ε$_{p}$ ) 〈 n | l$_{j}$ | l 〉 ( − → s N × − → s N − $_{1}$) | j 〈 l | r$_{k}$ | m 〉 Δ E$_{cf}$ + c.c.. (26)