Inequivalent Berry phases for the bulk polarization.

We address the widespread confusion in characterizing the polarization for insulators under the periodic boundary condition in terms of the Berry phase. The Berry phase can be formulated in terms of the Bloch function and the the lattice momentum for band insulators. More generally, in the presence of interactions or disorders, one can instead use the many-body ground state as a function of the flux piercing the ring. However, the definition of the Bloch function and the way describing the flux are not unique. As a result, the value of the polarization and its behavior depend on how precisely it is defined. We demonstrate this by formulating different definitions of the polarization in terms of Berry phases, and elucidating their mutual relations. The total charge transported during a Thouless pumping process is independent of the definition, reflecting its topological nature.