  In this paper, we propose a unified formalism, using Green's functions, to
integrate out the electrons in an insulator under uniform electromagnetic
fields. We derive a perturbative formula for the Green's function in the
presence of uniform magnetic or electric fields. Applying the formula, we
derive the formula for the polarization, the orbital magnetization, and the
orbital magneto-polarizability, without assuming time reversal symmetry.
Specifically, we realize that the terms linear in the electric field can only
be expressed in terms of the Green's functions in one extra dimension. This
observation directly leads to the result that the coefficient of the $\theta$
term in any dimensions is given by a Wess-Zumino-Witten-type term, integrated
in the extended space, interpolating between the original physical Brillouin
zone and a trivial system, with the group element replaced by the Green's
function. This generalizes an earlier result for the case of time reversal
invariance [see Z. Wang, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. {\bf 105},
256803 (2010)].

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