Magnetically induced polarization in centrosymmetric bonds

We reveal the microscopic origin of electric polarization $\vec{P}$ induced by noncollinear magnetic order. We show that in Mott insulators, such $\vec{P}$ is given by all possible combinations of position operators $\hat{\vec{r}}_{ij} = (\vec{r}_{ij}^{\, 0},\vec{\boldsymbol{r}}_{ij}^{\phantom{0}})$ and transfer integrals $\hat{t}_{ij} = (t_{ij}^{0},\boldsymbol{t}_{ij}^{\phantom{0}})$ in the bonds, where $\vec{r}_{ij}^{\, 0}$ and $t_{ij}^{0}$ are spin-independent contributions in the basis of Kramers doublet states, while $\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$ and $\boldsymbol{t}_{ij}^{\phantom{0}}$ stem solely from the spin-orbit interaction. Among them, the combination $t_{ij}^{0} \vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$, which couples to the spin current, remains finite in the centrosymmetric bonds, thus yielding finite $\vec{P}$ in the case of noncollinear arrangement of spins. The form of the magnetoelectric coupling, which is controlled by $\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$, appears to be rich and is not limited to the phenomenological law $\vec{P} \sim \boldsymbol{\epsilon}_{ij} \times [\boldsymbol{e}_{i} \times \boldsymbol{e}_{j}]$ with $\boldsymbol{\epsilon}_{ij}$ being the bond vector connecting the spins $\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{j}$. Using density-functional theory, we illustrate how the proposed mechanism work in the spiral magnets CuCl$_2$, CuBr$_2$, CuO, and $\alpha$-Li$_2$IrO$_3$, providing consistent explanation to available experimental data.