Berry Phase Effects in Dipole Density and the Mott Relation

We provide a unified semiclassical theory for thermoelectric responses of any observable represented by an operator $\widehat{\mathit{\theta }}$ that is well defined in periodic crystals. The Einstein and Mott relations are established generally in the presence of Berry phase effects for various physical realizations of $\widehat{\mathit{\theta }}$ in electronic systems, including the familiar case of the electric current as well as the currently controversial cases of the spin polarization and spin current. The magnetization current, which has been proven indispensable in the thermoelectric response of electric current, is generalized to the cases of various $\widehat{\mathit{\theta }}$. In our theory the dipole density of a physical quantity emerges and plays a vital role, which contains not only the statistical sum of the dipole moment of $\widehat{\mathit{\theta }}$ but also a Berry phase correction.

Figure 1

A schematic picture of $m^{i\mathit{\theta }}$, which is proportional to the difference between the $\mathit{\theta }$ center ${\mathit{r}}^{\mathit{\theta }}$ and the usual probability center ${\mathit{r}}_c$ (the red arrow), where the $\mathit{\theta }$ center is defined as ${\mathit{r}}^{\mathit{\theta }}\equiv (⟨\mathrm{\Phi }|\widehat{\mathit{r}}\widehat{\mathit{\theta }}|\mathrm{\Phi }⟩/⟨\mathrm{\Phi }|\widehat{\mathit{\theta }}|\mathrm{\Phi }⟩)$. By definition (3), $m^{i\mathit{\theta }}=⟨\mathrm{\Phi }|\widehat{\mathit{\theta }}|\mathrm{\Phi }⟩({\mathit{r}}^{\mathit{\theta }}-{\mathit{r}}_c{)}^i$. For “conserved” $\widehat{\mathit{\theta }}$ that commutes with the Hamiltonian ${\widehat{H}}_0$, the $\mathit{\theta }$ center coincides with the probability center such that the $\mathit{\theta }$ dipole moment vanishes.