Z$_2$ topological classification of the $S=1/2$ Heisenberg model on the two dimensional pylochlore lattice

We have investigated the $S=1/2$ Heisenberg model on the two dimensional pyrochlore lattice [1] by using the Z$_2$ Berry phase [2] which is quantized due to the time reversal invariance. Since these Z$_2$ Berry phases are adiabatic invariants against change of the physical parameters, one can distinguish ground states when the quantized values are different. By suitably choosing ways of local $U(1)$ twists to define the Berry phases, one can show that the ground state in this system is adiabatically connected to a direct product state of local singlets or plaquette singlets. We have described the basic formulation and demonstrate the validity for finite systems using a Lanczos diagonalization. \\[4pt] [1] J.-B. Fouet, M. Mambrini, P. Sindzingre, and C.Lhuillier, Phys. Rev. B 67 (2003) 054411.\\[0pt] [2] Y. Hatsugai, J.Phys Soc. Jpn. 73 (2004) 2604.