Z$_Q$ topological invariants for identification of short rangeentangled states

Since the Berry phase is quantized as $Z_2$ value, i.e., 0 or $\pi$, due to the time-reversal, or lattice-inversion symmetry in any dimension, the quantized Berry phase[1] is useful for characterization of a topological or quantum order in various models including strongly correlated electron systems[2] and spin systems[3]. Recently, we have proved $Z_Q$ quantization of Berry phases for the general lattice symmetry, where $Z_{Q}$ ($Q=d+1$) Berry phases are defined for $d$-dimensional lattices: Polyacetylene, Kagome and Pyrochlore lattice respectively for $d=1,2$ and $3$.[4]. We have also characterized the dimer-plaquette transition of the orthogonal dimer model in two dimension[5]. [1]{Y.Hatsugai, J. Phys. Soc. Jpn, {\bf 75}, 123601 (2006).} [2]{IM, Y. Hatsugai , J. Phys. Soc. Jpn, {\bf 76}, 113601 (2007).} [3]{IM, T.Hirano, Y.Hatsugai, Phys. Rev. B. {\bf 79}, 115107 (2009)}[4]{Y.Hatsugai, IM, Euro. Phys. Lett., {\bf 95}, 20003 (2011) }[5] I.Maruyama, S.Tanaya, M.Arikawa, Y.Hatsugai, J. Phys.:Conf. Ser., {\bf 320}, 01219 (2011)