Berry phases and finite electric fields

Berry phase expression for the macroscopic polarization

Calculating the change in dipole moment per unit cell under PBC's, is a nontrivial task. In general one {\it cannot} define it as the first moment of the induced change in charge density δ(r), through

[math]\displaystyle{ \Delta \mathbf{P}= \frac{1}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta \left( \mathbf{r} \right) d^{3}r }[/math]

without introducing a dependency on the shape of Ω₀, the chosen unit cell (see for instance Ref.\ \cite{Vogl78}).

Recently King-Smith and Vanderbilt\ \cite{Vanderbilt93I}, building on the work of Resta\ \cite{Resta92}, showed that the electronic contribution to the difference in polarization $\Delta \mathbf{P}_{e}$, due to a finite adiabatic change in the Hamiltonian of a system, can be identified as a {\it geometric quantum phase} or {\it Berry phase} of the valence wave functions. We will briefly summarize the essential results (for a review of geometric quantum phases in polarization theory see Refs.\ \cite{Resta94} and \cite{Resta96}).

Related Tags and Sections

LBERRY, IGPAR, NPPSTR, LPEAD, IPEAD, LCALCPOL, LCALCEPS, EFIELD_PEAD, DIPOL

References

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