  The behavior of electric polarization is related to intrinsic symmetry
properties of position operator $\vec{r}$ in the basis of Kramers doublet
states: for each bond, the $2$$\times$$2$ matrix $\hat{\vec{r}}_{ij} =
\vec{r}_{ij}^{\, 0} \hat{\mathbb{1}} + i
\vec{\boldsymbol{r}}_{ij}^{\phantom{0}} \hat{\boldsymbol{\sigma}}$ is
decomposed in terms of the unity $\hat{\mathbb{1}}$ and Pauli matrices
$\hat{\boldsymbol{\sigma}} =
(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ with the real vector
coefficients $(\vec{r}_{ij}^{\, 0},\vec{\boldsymbol{r}}_{ij}^{\phantom{0}})$,
reflecting the fact that $\vec{r}$ is spinless. In the Mott insulating state of
multiferroics, the polarization $\vec{P}$ is given by the all possible products
of $(\vec{r}_{ij}^{\, 0},\vec{\boldsymbol{r}}_{ij}^{\phantom{0}})$ and the
scalar coefficients $(t_{ij}^{0},\boldsymbol{t}_{ij}^{\phantom{0}})$ obtained
from a similar decomposition of the transfer integrals $\hat{t}_{ij} =
t_{ij}^{0} \hat{\mathbb{1}} + i \boldsymbol{t}_{ij}^{\phantom{0}}
\hat{\boldsymbol{\sigma}}$. Among them, the antisymmetric product $t_{ij}^{0}
\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$, which couples to the spin current,
remains finite even if the bond is centrosymmetric, thus permitting finite
$\vec{P}$ for noncollinear spins. The relative orientation of the bond vector
($\boldsymbol{\epsilon}_{ij}$), spin vectors ($\boldsymbol{e}_{i}$ and
$\boldsymbol{e}_{j}$), and $\vec{P}$ depends on the symmetry of the Kramers
states and is not limited to the phenomenological law $\vec{P} \sim
\boldsymbol{\epsilon}_{ij} \times [\boldsymbol{e}_{i} \times
\boldsymbol{e}_{j}]$. The proposed generalized spin-current theory is
successfully applied to spiral magnets CuCl$_2$, CuBr$_2$, CuO, and
$\alpha$-Li$_2$IrO$_3$.
