Quantized Berry Phases of a Spin-1/2 Frustrated Two-Leg Ladder with   Four-Spin Exchange
  A spin-1/2 frustrated two-leg ladder with four-spin exchange interaction is studied by quantized Berry phases. We found that the Berry phase successfully characterizes the Haldane phase in addition to the rung-singlet phase, and the dominant vector-chirality phase. The Hamiltonian of the Haldane phase is topologically identical to the S=1 antiferromagnetic Heisenberg chain. Decoupled models connected to the dominant vector-chirality phase revealed that the local object identified by the non-trivial ($\pi$) Berry phase is the direct product of two diagonal singlets. 
I. Maruyama, T. Hirano, and Y. Hatsugai