Z_N Berry phase and symmetry protected topological phases of SU(N) antiferromagnetic Heisenberg chain

The local Z_N quantized Berry phase for the SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local Z_2 Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning on a particular bond, and is effective as a tool to characterize and classify the various symmetry protected topological phases of one-dimensional SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the Z_2 Berry phase in order to calculate the ZN Berry phase numerically. We demonstrate our method by calculating the Z_4 Berry phase for the bond-alternating SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.