Microscopic theory of Chern polarization via crystalline defect charge

The modern theory of polarization does not apply in its original form to systems with non-trivial band topology. Chern insulators are one such example. Defining polarization for them is complicated because they are insulating in the bulk but exhibit metallic edge states. Wannier functions formed a key ingredient of the original modern theory of polarization, but it has been considered that these cannot be applied to Chern insulators since they are no longer exponentially localized and the Wannier center, obtained from the Zak phase, is no longer gauge invariant. In this article, we provide an unambiguous definition of absolute polarization for a Chern insulator in terms of the Zak phase. We obtain our expression by studying the non-quantized fractional charge bound to lattice dislocations. Our expression can be computed directly from bulk quantities and makes no assumption on the edge state filling. It is fully consistent with previous results on the quantized charge bound to dislocations in the presence of crystalline symmetry. At the same time, our result is more general since it also applies to Chern insulators which do not have crystalline symmetries other than translations.

💡 Research Summary

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The paper tackles a long‑standing problem in the modern theory of polarization: how to define an absolute bulk polarization for Chern insulators, whose non‑trivial band topology (non‑zero Chern number) prevents the construction of exponentially localized Wannier functions and renders the Zak phase gauge‑dependent. Traditional approaches either compute polarization changes during adiabatic deformations or rely on the filling of metallic edge states, but neither yields a unique, bulk‑only definition of polarization for a Chern insulator.

The authors begin by revisiting the surface‑charge theorem in one and two dimensions. In 1D, they show that the edge charge can be decomposed into an integer contribution (the difference between the numbers of ions and Wannier charge centers) and a fractional part proportional to the bulk Wannier center, i.e. the polarization. Extending to 2D via dimensional reduction, they treat each transverse momentum (k_y) as a separate 1D problem. For a trivial insulator the Wannier center returns to its original position after a full (k_y) cycle, while for a Chern insulator it winds by an integer number of unit cells. This winding forces charge to be transferred from one edge to the opposite edge, reflecting the presence of chiral edge modes.

To obtain a gauge‑invariant bulk quantity, the authors consider a lattice dislocation (a Burgers vector (\mathbf{G})). By inserting a dislocation they break translational symmetry locally but preserve it globally. Using only bulk Bloch wavefunctions they compute the bound charge (Q_D) localized at the dislocation. The result is a simple, fully bulk expression:

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