A Wannier Function Approach to Absolute Polarization for a 2D Chern Insulator

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 since they are insulating in the bulk but exhibit metallic edge states, complicating the definition of polarization. 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 is no longer gauge invariant. Our work aims to show that an extension of the modern theory of polarization for a Chern insulator in terms of Wannier functions is in fact possible. In PRR 6.023046, we presented a method of gauge-fixing to determine optimally localized Wannier functions for such systems. We derived the change in polarization under adiabatic deformations, in terms of Wannier functions, consistent with existing work. In the present work, we provide an unambiguous definition of absolute polarization for a Chern insulator using a Wannier prescription. Our expression can be computed directly from bulk quantities and makes no assumption on the edge state filling. We confirm our definition by studying fractional charges bound to lattice dislocations. Our result is fully consistent with previous results on the quantized charge bound to dislocations in the presence of crystalline symmetries. At the same time, our result is more general since it also applies to systems which do not have these symmetries.