Z2 topological number of local quantum clusters in the orthogonal dimer model

We have studied the Z2 topological number defined by the Berry phase for the gapped frustrated systems including the orthogonal dimer model which has a direct product state of local quantum clusters as the exact ground state. The Z2 topological number can clarify what kind of the local quantum clusters is formed to lift the macroscopic degeneracy due to frustration, even when the exact ground state is unknown. As a demonstration, the dimer-singlet and plaquette-singlet phase are identified by two kinds of Z2 topological numbers in the Shastry-Sutherland model and its generalization realized experimentally as SrCu2(BO3)2 and CaV4O9.