Electric polarization in Chern insulators: Unifying many-body and single-particle approaches

Recently, it has been established that Chern insulators possess an intrinsic two-dimensional electric polarization, despite having gapless edge states and non-localizable Wannier orbitals. This polarization, P→o, can be defined in a many-body setting from various physical quantities, including dislocation charges, boundary charge distributions, and linear momentum. Importantly, there is a dependence on a choice of real-space origin o within the unit cell. In contrast, Coh and Vanderbilt extended the single-particle Berry phase definition of polarization to Chern insulators by choosing an arbitrary point in momentum space, k→0. In this paper, we unify these two approaches and show that when the real-space origin o and momentum-space point k→0 are appropriately chosen in relation to each other, the Berry phase and many-body definitions of polarization are equal.