@article{oai:tsukuba.repo.nii.ac.jp:00027498,
 author = {初貝, 安弘 and Hatsugai, Y. and Maruyama, I.},
 issue = {2},
 journal = {EPL},
 month = {Jul},
 note = {Adiabatic ZQ 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 ZQ equivalence even with particle-particle interaction. As for the chiral symmetric case, a topological form of the Z2 invariant is explicitly given as well.},
 title = {ZQ topological invariants for Polyacetylene, Kagome and Pyrochlore lattices},
 volume = {95},
 year = {2011}
}