\begin{thebibliography}{35}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Bethe and Salpeter}(1957)}]{bethe_salpeter}
\bibinfo{author}{\bibfnamefont{H.~A.} \bibnamefont{Bethe}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{E.~E.} \bibnamefont{Salpeter}},
  \emph{\bibinfo{title}{Quantum Mechanics of One- and Two-Electron Atoms}}
  (\bibinfo{publisher}{Academic Press}, \bibinfo{year}{1957}).

\bibitem[{\citenamefont{Hasegawa and Howard}(1961)}]{hasegawa61}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Hasegawa}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~E.} \bibnamefont{Howard}},
  \bibinfo{journal}{J. Phys. Chem. Solids} \textbf{\bibinfo{volume}{21}},
  \bibinfo{pages}{179} (\bibinfo{year}{1961}).

\bibitem[{\citenamefont{Smith}(1976)}]{smith76}
\bibinfo{author}{\bibfnamefont{D.~Y.} \bibnamefont{Smith}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{13}},
  \bibinfo{pages}{5303} (\bibinfo{year}{1976}).

\bibitem[{\citenamefont{Thole et~al.}(1992)\citenamefont{Thole, Carra, Sette,
  and van~der Laan}}]{thole92}
\bibinfo{author}{\bibfnamefont{B.~T.} \bibnamefont{Thole}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Carra}},
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Sette}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{van~der Laan}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{68}},
  \bibinfo{pages}{1943} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Oppeneer}(1998)}]{oppeneer98}
\bibinfo{author}{\bibfnamefont{P.~M.} \bibnamefont{Oppeneer}},
  \bibinfo{journal}{J. Magn. Magn. Mat.} \textbf{\bibinfo{volume}{188}},
  \bibinfo{pages}{275} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Thonhauser et~al.}(2005)\citenamefont{Thonhauser,
  Ceresoli, Vanderbilt, and Resta}}]{timo05}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Thonhauser}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ceresoli}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Resta}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{95}},
  \bibinfo{pages}{137205} (\bibinfo{year}{2005}).

\bibitem[{\citenamefont{Xiao et~al.}(2005)\citenamefont{Xiao, Shi, and
  Niu}}]{xiao05}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Xiao}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Shi}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{95}},
  \bibinfo{pages}{137204} (\bibinfo{year}{2005}).

\bibitem[{\citenamefont{Ceresoli et~al.}(2006)\citenamefont{Ceresoli,
  Thonhauser, Vanderbilt, and Resta}}]{ceresoli06}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ceresoli}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Thonhauser}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Resta}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{74}},
  \bibinfo{pages}{024408} (\bibinfo{year}{2006}).

\bibitem[{\citenamefont{Shi et~al.}(2007)\citenamefont{Shi, Vignale, Xiao, and
  Niu}}]{shi07}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Shi}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Vignale}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Xiao}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}},
  \bibinfo{journal}{arXiv:0704.3824}  (\bibinfo{year}{2007}).

\bibitem[{\citenamefont{Haldane}(1988)}]{haldane88}
\bibinfo{author}{\bibfnamefont{F.~D.~M.} \bibnamefont{Haldane}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{2015} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Bennett and Stern}(1965)}]{bennett65}
\bibinfo{author}{\bibfnamefont{H.~S.} \bibnamefont{Bennett}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{E.~A.} \bibnamefont{Stern}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{137}},
  \bibinfo{pages}{A448} (\bibinfo{year}{1965}).

\bibitem[{\citenamefont{Kittel}(1953)}]{kittel53}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Kittel}},
  \emph{\bibinfo{title}{Introduction to Solid State Physics}}
  (\bibinfo{publisher}{Wiley}, \bibinfo{year}{1953}), \bibinfo{edition}{1st}
  ed.

\bibitem[{\citenamefont{Marzari and Vanderbilt}(1997)}]{mv97}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Marzari}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{56}},
  \bibinfo{pages}{12847} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Thonhauser and Vanderbilt}(2006)}]{timo06}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Thonhauser}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{74}},
  \bibinfo{pages}{235111} (\bibinfo{year}{2006}).

\bibitem[{exp({\natexlab{a}})}]{explan-notation}
\bibinfo{note}{The quantity $\sr$ was called ${\bf M}_{\rm LC}$ in
  Ref.~\onlinecite{timo05}. One reason for using a different notation here is
  that in Ref.~\onlinecite{ceresoli06} ${\bf M}_{\rm LC}$ was defined (for
  crystals) in a different way, by replacing the Wannier center $\overline\r_i$
  in Eq.~(\ref{eq:Msr}) by the lattice vector $\R$ labeling the cell to which
  the WF ``belongs.'' The two definitions turn out to be equivalent for
  single-band insulators (the case of interest in Ref.~\onlinecite{timo05}). In
  Sec.~\ref{sec:relation_morb_bulk} we discuss the precise relation between our
  $\sr$ and the ${\bf M}_{\rm LC}$ of Ref.~\onlinecite{ceresoli06} in the
  multiband case.}

\bibitem[{\citenamefont{Souza et~al.}(2000)\citenamefont{Souza, Wilkens, and
  Martin}}]{souza00}
\bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Souza}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Wilkens}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~M.} \bibnamefont{Martin}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{62}},
  \bibinfo{pages}{1666} (\bibinfo{year}{2000}).

\bibitem[{exp({\natexlab{b}})}]{explan-offdiag}
\bibinfo{note}{However, since the off-diagonal $\r_{ij}$ enter the
  self-rotation as a first power and the spread as a second power, the
  non-invariant part may play a larger role in the former quantity than in the
  latter.}

\bibitem[{\citenamefont{King-Smith and Vanderbilt}(1993)}]{ksv93}
\bibinfo{author}{\bibfnamefont{R.~D.} \bibnamefont{King-Smith}}
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{47}},
  \bibinfo{pages}{1651} (\bibinfo{year}{1993}).

\bibitem[{\citenamefont{Souza et~al.}(2004)\citenamefont{Souza,
  {\'I}{\~n}iguez, and Vanderbilt}}]{souza04}
\bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Souza}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{{\'I}{\~n}iguez}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{69}},
  \bibinfo{pages}{085106} (\bibinfo{year}{2004}).

\bibitem[{exp({\natexlab{c}})}]{explan-intra}
\bibinfo{note}{For metals the low-frequency part of the absorption spectrum
  contains intraband (Drude) contributions in addition to the interband
  absorption. Only the latter was taken into account in the derivation of the
  dichroic $f$-sum rule.}

\bibitem[{\citenamefont{Xiao et~al.}(2006)\citenamefont{Xiao, Yao, Fang, and
  Niu}}]{xiao06}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Xiao}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Yao}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{97}},
  \bibinfo{pages}{026603} (\bibinfo{year}{2006}).

\bibitem[{\citenamefont{Vanderbilt and King-Smith}(1993)}]{vks93}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~D.} \bibnamefont{King-Smith}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{48}},
  \bibinfo{pages}{4442} (\bibinfo{year}{1993}).

\bibitem[{exp({\natexlab{d}})}]{explan-quantum}
\bibinfo{note}{The quantum of polarization can ultimately be traced to the
  quantization of the charge in a Wannier function. On the contrary, neither
  the self-rotation nor the center-of-mass velocity of a Wannier function is
  quantized; these can change continuously under a multiband gauge
  transformation.}

\bibitem[{\citenamefont{Yafet}(1963)}]{yafet63}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Yafet}},
  \bibinfo{journal}{Solid State Phys.} \textbf{\bibinfo{volume}{14}},
  \bibinfo{pages}{1} (\bibinfo{year}{1963}).

\bibitem[{\citenamefont{Chen et~al.}(1995)\citenamefont{Chen, Idzerda, Lin,
  Smith, Meigs, Chaban, Ho, Pellegrin, and Sette}}]{chen95}
\bibinfo{author}{\bibfnamefont{C.~T.} \bibnamefont{Chen}},
  \bibinfo{author}{\bibfnamefont{Y.~U.} \bibnamefont{Idzerda}},
  \bibinfo{author}{\bibfnamefont{H.-J.} \bibnamefont{Lin}},
  \bibinfo{author}{\bibfnamefont{N.~V.} \bibnamefont{Smith}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Meigs}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Chaban}},
  \bibinfo{author}{\bibfnamefont{G.~H.} \bibnamefont{Ho}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Pellegrin}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Sette}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{75}},
  \bibinfo{pages}{152} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Hjelm et~al.}(1995)\citenamefont{Hjelm, Trygg,
  Eriksson, Johansson, and Wills}}]{hjelm95}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Hjelm}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Trygg}},
  \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Eriksson}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Johansson}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.~M.} \bibnamefont{Wills}},
  \bibinfo{journal}{Int. J. Mod. Phys. B} \textbf{\bibinfo{volume}{9}},
  \bibinfo{pages}{2735} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Yaresko et~al.}(1998)\citenamefont{Yaresko, Uba, Uba,
  Perlov, Gontarz, and Antonov}}]{yaresko98}
\bibinfo{author}{\bibfnamefont{A.~N.} \bibnamefont{Yaresko}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Uba}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Uba}},
  \bibinfo{author}{\bibfnamefont{A.~Y.} \bibnamefont{Perlov}},
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Gontarz}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~N.} \bibnamefont{Antonov}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{58}},
  \bibinfo{pages}{7648} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Ebert and Man'kovsky}(2003)}]{ebert03}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Ebert}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Man'kovsky}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{90}},
  \bibinfo{pages}{077404} (\bibinfo{year}{2003}).

\bibitem[{\citenamefont{Yates et~al.}()\citenamefont{Yates, Vanderbilt, and
  Souza}}]{yates-unpublished}
\bibinfo{author}{\bibfnamefont{J.~R.} \bibnamefont{Yates}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vanderbilt}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Souza}},
  \bibinfo{note}{(in preparation).}

\bibitem[{\citenamefont{Mott and Jones}(1936)}]{mott_jones}
\bibinfo{author}{\bibfnamefont{N.~F.} \bibnamefont{Mott}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Jones}},
  \emph{\bibinfo{title}{The Theory of the Properties of Metals and Alloys}}
  (\bibinfo{publisher}{Clarendon Press}, \bibinfo{year}{1936}).

\bibitem[{\citenamefont{Yao et~al.}(2004)\citenamefont{Yao, Kleinman,
  MacDonald, Sinova, Jungwirth, Wang, Wang, and Niu}}]{yao04}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Yao}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Kleinman}},
  \bibinfo{author}{\bibfnamefont{A.~H.} \bibnamefont{MacDonald}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Sinova}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Jungwirth}},
  \bibinfo{author}{\bibfnamefont{D.-S.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Wang}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{92}},
  \bibinfo{pages}{037204} (\bibinfo{year}{2004}).

\bibitem[{\citenamefont{Resta}(2002)}]{resta02}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Resta}}, \bibinfo{journal}{J.
  Phys.: Condens. Matter} \textbf{\bibinfo{volume}{14}}, \bibinfo{pages}{R625}
  (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Kohn}(1968)}]{kohn68}
\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Kohn}}, in
  \emph{\bibinfo{booktitle}{Many-Body Physics}}, edited by
  \bibinfo{editor}{\bibfnamefont{C.}~\bibnamefont{DeWitt}} \bibnamefont{and}
  \bibinfo{editor}{\bibfnamefont{R.}~\bibnamefont{Balian}}
  (\bibinfo{publisher}{Gordon and Breach}, \bibinfo{address}{New York},
  \bibinfo{year}{1968}), p. \bibinfo{pages}{351}.

\bibitem[{exp({\natexlab{e}})}]{explan-vanishing}
\bibinfo{note}{To see that the middle expression in Eq.~(\ref{eq:kuhn})
  vanishes, write ${\rm Im}\,{\rm Tr}[\P\hat r_\alpha\Q\hat r_\beta]$ as ${\rm
  Im}\,{\rm Tr}[(\P)(\hat r_\alpha\hat r_\beta)]- {\rm Im}\,{\rm Tr}[(\P\hat
  r_\alpha\P)(\P\hat r_\beta\P)]$, note that $\P$, $\hat r_\alpha\hat r_\beta$,
  $\P\hat r_\alpha\P$ and $\P\hat r_\beta\P$ are all Hermitian, and use
  Eq.~(\ref{eq:rule3}).}

\bibitem[{\citenamefont{Kunes and Oppeneer}(2000)}]{kunes00}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kunes}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{P.~M.} \bibnamefont{Oppeneer}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{15774} (\bibinfo{year}{2000}).

\end{thebibliography}
