Topological identification of a spin-1/2 two-leg ladder with four-spin ring exchange

A spin-1/2 two-leg ladder with four-spin ring exchange is studied by quantized Berry phases, used as local-order parameters. Reflecting local objects, nontrivial (pi) Berry phase is founded on a rung for the rung-singlet phase and on a plaquette for the vector-chiral phase. Since the quantized Berry phase is topologically invariant for gapped systems with the time-reversal symmetry, topologically identical models can be obtained by the adiabatic modification. The rung-singlet phase is adiabatically connected to a decoupled rung-singlet model and the vector-chiral phase is connected to a decoupled vector-chiral model. Decoupled models reveal that the local objects are a local singlet and a plaquette singlet, respectively.