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

Theory of electric polarization induced by inhomogeneity in crystals