Sequential Topological Phase Transitions of J₁-J₂ Integer Spin Chain

Phase diagrams of the integer spin Heisenberg chains with nearest neighbor (J₁) and next nearest neighbor (J₂) interaction are considered numerically for S = 1, 2, 3 and 4. The Berry phase associated with twisting several local bonds is quantized to 0 or π (Z₂) due to the time reversal symmetry, which is a topological order parameter assuming the gap is finite. Since it is an adiabatic invariant even for a finite system, the phase boundaries are well defined for the finite system. This Z₂ Berry phase is a topological order parameter for the short range Symmetry Protected Topological (SPT) phases. Several Z₂ Berry phases are evaluated by choosing a combination of local bonds. Sequential quantum phase transitions by changing J₂/J₁ are observed which implies reconstruction of local singlets.