  We show that the $Z_N$ Berry phase (Berry phase quantized into $2\pi/N$)
provides a useful tool to characterize symmetry protected topological phases
with correlation that can be directly computed through numerics of a relatively
small system size. The $Z_N$ Berry phase is defined in a $N-1$ dimensional
parameter space of local gauge twists, which we call "synthetic Brillouin
zone", and an appropriate choice of an integration path consistent with the
symmetry of the system ensures exact quantization of the Berry phase. We
demonstrate the usefulness of the $Z_N$ Berry phase by studying two 1D models
of bosons, SU(3) and SU(4) AKLT models, where topological phase transitions are
captured by $Z_3$ and $Z_4$ Berry phases, respectively. we find that the exact
quantization of the $Z_N$ Berry phase at the topological transitions arises
from a gapless band structure (e.g., Dirac cones or nodal lines) in the
synthetic Brillouin zone.
