content_image/295179.jpg content_image/295173.jpg content_image/295175.jpg content_image/295176.jpg content_image/295177.jpg content_image/295178.jpg
into eq. (24), we found that 

<img src='content_image/295179.jpg'>

The connection between the electric polarization and spin current can now be established. A common definition of the spin current is 

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 

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 

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