Berry Phases and Curvatures in Electronic-Structure Theory.

In the last fifteen years, Berry phases have been found to play an increasingly important role in electronic-structure theory. I will briefly review some of the important developments in which Berry phases have been involved, starting with the modern theory of polarization$^1$ and the closely related theory of Wannier functions and their Wannier centers.$^2$ Next, I will discuss the theory of insulators in finite electric fields,$^3$ in which the field is taken to couple linearly to the Berry-phase polarization. I will then conclude by discussing the role of Berry phases and Berry curvatures in systems in which time-reversal symmetry has been broken, and in particular, the theory of orbital magnetization$^4$ and the anomalous Hall effect in ferromagnets. \begin{itemize} \item[{[1]}] R.D.~King-Smith and D.~Vanderbilt, Phys. Rev. B {\bf 47}, 1651 (1993). \item[{[2]}] Nicola Marzari and D.~Vanderbilt, Phys. Rev. B {\bf 56}, 12847 (1997). \item[{[3]}] I.~Souza, J.~\'I\~niguez, and D.~Vanderbilt, Phys. Rev. Lett. {\bf 89}, 117602 (2002). \item[{[4]}] T.~Thonhauser, D.~Ceresoli, D.~Vanderbilt, and R.~Resta, Phys. Rev. Lett. {\bf 95}, 137205 (2005). \end{itemize}