Recently, it has been established that Chern insulators possess an intrinsic two-dimensional electric polarization, despite having gapless edge states and nonlocalizable Wannier orbitals. This polarization, ${\vec{P}}_{\text{o}}$, 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, ${\vec{k}}_0$. In this paper, we unify these two approaches and show that when the real-space origin o and momentum-space point ${\vec{k}}_0$ are appropriately chosen in relation to each other, the Berry phase and many-body definitions of polarization are equal.

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• Chern insulators
• Electric polarization
• Fractional quantum Hall effect
• Geometric & topological phases
• Quantum Monte Carlo