Berry phases and finite electric fields: Difference between revisions
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== Berry phase expression for the macroscopic polarization == | == 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> | |||
\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 Ω<sub>0</sub>, 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}). | |||
== Self-consistent response to finite electric fields == | == Self-consistent response to finite electric fields == | ||
Revision as of 14:01, 6 March 2011
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
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}).
Self-consistent response to finite electric fields
Related Tags and Sections
LBERRY, IGPAR, NPPSTR, LPEAD, IPEAD, LCALCPOL, LCALCEPS, EFIELD_PEAD, DIPOL