  We develop a general theory of electric polarization induced by inhomogeneity
in crystals. We show that contributions to polarization can be classified in
powers of the gradient of the order parameter. The zeroth order contribution
reduces to the well-known result obtained by King-Smith and Vanderbilt for
uniform systems. The first order contribution, when expressed in a two-point
formula, takes the Chern-Simons 3-form of the vector potentials derived from
the Bloch wave functions. Using the relation between polarization and charge
density, we demonstrate our formula by studying charge fractionalization in a
two-dimensional dimer model recently proposed.
