Z_Q Topological Invariants for Polyacetylene, Kagome and Pyrochlore lattices

Adiabatic $Z_Q$ invariants by quantized Berry phases are defined for gapped electronic systems in $d$-dimensions ($Q=d+1$). This series includes Polyacetylene, Kagome and Pyrochlore lattice respectively for $d=1,2$ and 3. The invariants are quantum $Q$-multimer order parameters to characterize the topological phase transitions by the multimerization. This fractional quantization is protected by the global $Z_Q$ equivalence. As for the chiral symmetric case, a topological form of the $Z_2$-invariant is explicitly given as well.