  We discuss characterization of the polarization for insulators under the
periodic boundary condition in terms of the Berry phase, clarifying confusing
subtleties. For band insulators, the Berry phase can be formulated in terms of
the Bloch function in the momentum space. 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 Berry phase and its behavior depend on how precisely it is defined. In
particular, identifying the Berry phase as a polarization, its change
represents a polarization current which also depends on the definition. We
demonstrate this by elucidating mutual relations among different definitions of
the Berry phase, and that they correspond to the current measured differently
in the real space. Despite the non-uniqueness of the polarization current, the
total charge transported during a Thouless pumping process is independent of
the definition, reflecting its topological nature.
